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A  HISTORY  OF 
THE  CONCEPTIONS  OF 
LIMITS  AND  FLUXIONS 
IN    GREAT    BRITAIN 

FROM 

NEWTON    IO   WOODHOUSE 


Copyright  in  Great  Britain  under  the  Ad  of  i<)ii 


TABLE    OF    CONTENTS 

INTRODUCTION 


PAGE 

I 


CHAPTER    I 
Newton 

Newton's  Publications  printed  before  1734  .         .         .         ,         .  2 

Principia     ..........  3 

Wallis's  De  Algebra  Tractatus      ......  14 

Quadratura  Curvaruin{\']0\) 17 

An  Account  of  the  Conwiercitim  Epistoliciim         ...  26 

Newton's  Correspondence  and  Manuscripts  not  in  print  in  1734  .  29 

Remarks     ...........  32 


CHAPTER     II 
Printed  Books  and  Articles  on  Fluxions 

BEFORE    1734 

John  Craig,  De  Moivre,  David  Gregory,  Patio  de  Duillier,  Cotes 

Ditton.  Cheyne    . 
John  Harris,  1702,  1705,  1710 
Charles  Hayts,  1704  . 
William  Jones,  1706   . 
Humphry  Ditton,  1706 

Commerciiim  Epistolicum  D.  Johannis  Collins,  17 12 
Joseph  Raphson,  17 15 
Brook  Taylor,  17 15     . 
James  Stirling,  17 17,  1730  . 
Edmund  Stone,  1730 
Remarks     .... 


37 
40 

41 
43 
43 
47 
49 
50 
50 
50 
55 


VI 


LIMITS    AND   FLUXIONS 


CHAPTER    III 

BERKELEY'S   ANALYST  (1734);     CONTROVERSY   VVITH 
JURIN    AND   WALTON 

The  Analyst        .... 
Jurin's  first  reply  to  Berkeley 
Walton's  first  reply  to  Berkeley  . 
Berkeley's  reply  to  Jurin  and  Walton 
Walton's  second  reply  to  Berkeley 
Jurin's  second  reply  to  Berkeley  . 
Berkeley's  second  reply  to  Walton 
The  second  edition  of  Walton's  second  reply 
Remai- Ics     ....... 


57 
64 
69 

72 

78 
80 

85 
87 
89 


CHAPTER    IV 

Jurin's  Controversy  with  Robins  and 
Pemberton 

Robin s's  Discourse  on  Fhixioiis  .......       96 

Jurin's  review  ofhis  own  letters  to  Berkeley          ....     loi 

Robins's  rejoinder        .  .  .  .  .  .  .  .  .106 

The  debate  continued  ........      109 

Pemberton  enters  the  debate       .          .         .         .         .         .         .129 

Debate  over   Robins's   review  of  treatises    by   Leonhard    Euler, 

Robert  Smith,  and  James  Jurin    .         .  .  .  .  .139 

Remar ks     .  .         .  .         .  .  .         .  .         .         .146 


CHAPTER    V 

TEXT-BOOKS   IMMEDIATELY    FOLLOWING 

Berkeley's  Attack 

John  Colson,  1736 149 

James  llodgson,  1736 155 

Thomas  Bayes,  1736  .         . 157 

John  Muller,  1736 162 

Anonymous  translation  of  Newton's  Methcd  oj  Fluxiovs,  1737      .  164 

James  Smith,  1737 165 

Thomas  Simpson,  1737        ........  169 

l')enjamin  Martin,  1739,  1759 171 

An  anonymous  text,  1741    ........  172 

John  Rowe,  1741,  1757,  1767 •  175 

Berkeley  ten  years  after       .  .  .  .  .         .         .         .178 

Remarks 179 


TABLE   OF   CONTENTS 


VII 


CHAPTER    VI 
Maclaurin's  "Treatise  OF  Fluxions,  1742" 


Remarks 


CHAPTER    VII 

TEXT-BOOKS  OF  THE   MIDDLE  OF  THE   CENTURY 


John  Stewart,  1745     .... 

William  Emerson,  1743  (?),  1757,  1768 

Thomas  Simpson,  1750 

Nicholas  Saunderson,  1756 

John  Rowning,  1756 

Israel  Lyons,  1758 

William  West,  1762 

James  Wilson,  1761 

Remarks     . 


190 
192 
194 
197 
198 
201 
202 
202 
206 


CHAPTER    Vili 

Robert  Heath  and  Friends  of  Emerson  in  Controversy 
wiTH  John  Turner  and  Friends  of  Simpson 


Robert  Heath 

Main  articles  in  the  controversy  . 
Ladies'  Diary^  1751,  1752. 
Popular  impression  of  the  nature  of  fluxions 
Remarks     ....... 


207 
209 
219 
222 
223 


CHAPTER    IX 
Abortive  Attempts  at  Arithmetisation 


John  Kirkby,  1748 

•     225 

John  Petvin,  1750 

.      230 

John  Landen,  1758     

.     231 

James  Glenie,  1793 

•      235 

Remarks     ......... 

.         .     238 

vili 


LIMirS    AND   FLUXIONS 


CHAPTER    X 
Later  Books  and  Articles  on  Fluxions 


Encyclopa:dia  Britannica^  iJTi,  i779,  1797 

Robert  Thorp,  1777   . 

F.  Holliday,  1777 

Charles  Hutton,  1796,  1798 

S.  Vince,  1795,  1805  . 

Agnesi — Colson — Hellins,  1801  . 

T.  Newton,  1805 

William  Dealtry,  1810,  18 16 

New  editions       .... 

Keviarks     ..... 


PAGE 

240 
241 

243 

244 

245 
247 
250 
252 
253 
253 


CHAPTER    XI 

Criticisms  of  Fluxions  by  British  Writers  under  the 
Influence  of  D'Alembert,  Lagrange,  and  Lacroix 


Review  of  Lagrange's  Fonctions  analyiigues, 
Review  of  a  memoir  of  Stockler,  1799 
Review  of  Lacroix's  Calcul  différentiel^  1800 
Review  of  Carnot's  Kéjlexions,  1801 
Robert  Woodhouse,  1803    . 
William  Hales,  1804  . 
Encyclopcedia  Britannica,  18 io  . 
Lacroix's  Elenientary  Treatise,  1816 
Remar ks     ..... 


1799 


255 
259 
260 
262 
263 
267 
269 
271 
274 


CHAPTER    XH 

Merits  and  Defects  of  the  Eighteenth-Century 
British  Fluxional  Conceptions 

Merits 277 

Defects       .         .  279 

Addenda 289 

Indkx 294 


LIMITS  AND  FLUXIONS 


INTRODUCTION 

I.  EVERV  great  epoch  in  the  progress  of  science  is 
preceded  by  a  period  of  preparation  and  prevision. 
The  invention  of  the  differential  and  integrai  calculus 
is  said  to  mark  a  "  crisis  "  in  the  history  of  mathe- 
matics.  The  conceptions  brought  into  action  at 
that  great  time  had  been  long  in  preparation.  The 
fluxional  idea  occurs  among  the  schoolmen — among 
Galileo,  Roberval,  Napier,  Barrow,  and  others.  The 
differences  or  differentials  of  Leibniz  are  found  in 
crude  form  among  Cavalieri,  Barrow,  and  others. 
The  undeveloped  notion  of  limits  is  contained  in 
the  ancient  method  of  exhaustion  ;  limits  are  found 
in  the  writings  of  Gregory  St.  Vincent  and  many 
others.  The  history  of  the  conceptions  which  led 
up  to  the  invention  of  the  calculus  is  so  extensive 
that  a  good-sized  volume  could  be  written  thereon. 
We  shall  not  yield  to  the  temptation  of  lingering 
on  this  pre-history  at  this  time,  but  shall  proceed 
at  once  to  the  subject-matter  of  the  present 
monograph. 


CHAPTER    I 

NEWTON 

2.  It  was  in  the  year  1734  that  Bishop  Berkeley 
made  his  famous  attack  upon  the  doctrine  of 
fluxions,  which  was  the  starting-point  of  ali  philo- 
sophical  discussion  of  the  new  mathematics  in 
England  during  the  eighteenth  century.  In  what 
foUows  we  quote  from  the  writings  of  Newton  that 
were  printed  before  1734  sudi  parts  as  bear  on  his 
conceptions  of  fluxions,  so  that  the  reader  may 
judge  for  himself  what  grounds  there  were  for 
Berkeley's  great  assault.  To  assist  us  in  the  inter- 
pretation  of  some  of  these  printed  passages,  we 
quote  also  from  manuscripts  and  letters  of  Newton 
which  at  that  time  were  stili  unprinted.  In  the 
next  chapter  we  give  an  account  of  the  foundations 
of  fluxions  as  displayed  by  other  writers  in  books 
and  articles  printed  in  Great  Britain  before  1734. 
It  is  hoped  that  the  material  contained  in  these  first 
two  chapters  will  enable  the  student  to  foUow  closely 
and  critically  the  debates  on  fluxions. 

Front  Neivton's  Publications  printed  before  1734 

I.   Principia 
3.    Three  editions  of  the  Principia  were  brought 
cut    in    Newton's    lifetime  ;    the   first    in    1687,    the 


NEWTON  3 

second  in  1713,  the  third  in  1726.  We  give  extracts 
which  bear  on  the  theory  of  limits  and  fluxions  and 
indicate  the  changes  in  phraseology  introduced  in 
the  second  and  third  editions.  We  give  also  trans- 
lations  into  EngUsh  based  on  the  text  of  the  1726, 
or  third,  edition. 

Principia,  Book  /,  Section  /,  Lemma  I 
First  edition  : 

4.  "  Quantitates,  ut  &  quantitatum  rationes,  quae 
ad  aequaUtatem  dato  tempore  constanter  tendunt 
&  eo  pacto  propius  ad  invicem  accedere  possunt 
quam  prò  data  quavis  differentia  ;  fiunt  ultimo 
aequales. 

5.  "Si  negas,  sit  earum  ultima  differentia  D. 
Ergo  nequeunt  propius  ad  a^qualitatem  accedere 
quam  prò  data  differentia  D  :  contra  hypothesin." 

Second  and  third  editions  : 

6.  "  Ouantitates,  ut  &  quantitatum  rationes,  quae 
ad  aequalitatem  tempore  quovis  finito  constanter 
tendunt,  &  ante  finem  temporis  illius  propius  ad 
invicem  accedunt  quam  prò  data  quavis  differentia, 
fiunt  ultimo  xquales. 

7.  "Si  negas,  fiant  ultimò  ina^quales,  &  sit 
earum,  etc."     [As  in  the  first  edition.] 

Translation  by  Robert  Thorp:^ 

8.  "Quantities,  and  the  ratios  of  quantities,  which, 
in  any  finite  time,  tend  continually  to  equality;  and 

^  Matheniatical  Principles  of  Naturai  Philosophy,  by  Sir  Isaac 
Newton,  Knight.  Trnnslated  into  Engiish,  and  illustrated  with  a 
Comtncntary,  by  Robert  Thorp,  M.A.,  voi.  i,  London,  1777. 


LIMITS    AND   FLUXIONS 


LF 


before  the  end  of  that  time,  approach  nearer  to  each 
other  than  by  any  given  difference,  become  ulti- 
mately  equal. 

"  If  you  deny  it,  let  them  be  ultimately  unequal  ; 
and  let  their  ultimate  difference  be  D.  Therefore, 
they  cannot  approach  nearer  to  equality  than  by 
that  given  difference  D.  Which  is  against  the 
supposition." 

Principia,  Book  /,  Sectiott  /,  Lemma  II 
Translation  by  Motte  :^ 

9.  ''If  in  any  figure  AacK,  terminated  by  the 
right  lines  Aa,  A  E,  and  the  curve  acY.,  there  be 
inscribed  any  number  of  paral- 
lelograms  Ab,  B<:,  Cd,  etc, 
comprehended  under  equal  bases 
AB,  BC,  CD,  etc,  and  the  sides 
^b.  Ce,  Dd,  etc,  parallel  to  one 
side  Aa  of  the  figure  ;  and  the 
parallelograms  aKb/,  bhejn, 
eMdn,  etc,  are  completed. 
Then  if  the  breadth  of  those 
parallelograms  be  supposed  to 
be  diminished,  and  their  number  be  augmented  in 
infinitum  ;  1  say,  that  the  ultimate  ratios  which 
the  inscribed  figure  AK<^L<:M<^D,  the  circumscribed 
figure  Aalbmcndo^.,  and  curvilinear  figure  AabedK, 
will  bave  to  one  another,  are  ratios  of  equality. 

^  T/ie  Mathematical  Principks  of  Naturai  Philosophy,  by  Sir  Isaac 
Newton;  transiated  into  English  ^j/ Andrew  Motte,  London,  1729. 
(Two  volumes.) 


II 

m 

b 

pN 

L 

a 

1    c 

M 

\ 

d 

\ 

BF    C 


NE  WTON  5 

**For  the  difference  of  the  inscribed  and  circum- 
scribed  figures  is  the  sum  of  the  parallelograms  K/, 
\^rn,  M«,  D(?,  that  is  (from  the  equality  of  ali  their 
bases),  the  rectangle  under  one  of  their  bases  K^ 
and  the  sum  of  their  altitudes  ha,  that  is,  the 
rectangle  AB/<a:.  But  this  rectangle,  because  its 
breadth  AB  is  supposed  diminished  in  infinitum, 
becomes  less  than  any  given  space.  And  therefore 
(by  Lem.  I)  the  figures  inscribed  and  circumscribed 
become  ultimately  equal  one  to  the  other  ;  and 
much  more  will  the  intermediate  curvilinear  figure 
be  ultimately  equal  to  either.      O.E.D." 

Principia,  Book  /,  Section  /,  Lemma  XI, 
Scholiuin  (first  part  omitted) 

IO.  ".  .  .  Quai  de  curvis  lineis  deque  super- 
ficiebus  comprehensis  demonstrata  sunt,  facile  appli- 
cantur.ad  solidorum  superficies  curvas  &  contenta. 
Praemisi  vero  haec  Lemmata,  ut  effugerem  tasdium 
deducendi  perplexas^  demonstrationes,  more  veterum 
Geometrarum,  ad  absurdum.  Contractiores  enim 
redduntur  demonstrationes  per  methodum  Indivisi- 
bilium.  Sed  quoniam  durior  est  Indivisibilium 
hypothesis,  &  propterea  methodus  illa  minus  Geo- 
metrica censetur  ;  malui  demonstrationes  rerum 
sequentium  ad  ultimas  quantitatum  evanescentium 
summas  &  rationes,  primasque  nascentium,  id  est, 
ad  limites  summarum  &  rationum  deducere  ;  & 
propterea  limitum  illorum  demonstrationes  qua  potui 
brevitate    praemittere.      His    enim    idem   prsestatur 

^  In  the  third  edition  "  longas"  takes  the  place  of  "  perplexas." 


6  LIMITS    AND   FLUXIONS 

quod  per  methodum  Indivisibilium  ;  &  principiis 
demonstratis  jam  tutius  utemur.  Proinde  in  sequen- 
tibus,  siquando  quantitates  tanquam  ex  particulis 
constantes  consideravero,  vel  si  prò  rectis  usurpavero 
lineolas  curvas  ;  nolim  indivisibilia,  sed  evanescentia 
divisibilia,  non  summas  &  rationes  partium  deter- 
minatarum,  sed  summarum  &  rationum  limites 
semper  intelligi  ;  vimque  talium  demonstrationum 
ad  methodum  praecedentium  Lemmatum  semper 
revocari. 

II.  "  Objectio  est,  quod  quantitatum  evanes- 
centium  nulla  sit  ultima  proportio  ;  quippe  qua^, 
antequam  evanuerunt,  non  est  ultima,  ubi  evanu- 
erunt,  nulla  est.  Sed  &  eodem  argumento  aeque 
contendi  posset  nullam  esse  corporis  ad  certum 
locum  pergentis  ^  velocitatem  ultimam.  Hanc  enim, 
antequam  corpus  attingit  locum,  non  esse  ultimam, 
ubi  attingit,  nullam  esse.  Et  responsio  facilis  est. 
Per  velocitatem  ultimam  intelligieam,^  qua  corpus 
movetur  ;  neque  antequam  attingit  locum  ultimum 
&  motus  cessat,  neque  postea,  sed  tunc  cum 
attingit  ;  id  est,  illam  ipsam  velocitatem  quacum 
corpus  attingit  locum  ultimum  &  quacum  motus 
cessat.  Et  similiter  per  ultimam  rationem  quanti- 
tatum evanescentium  intelligendam  esse  rationem 
quantitatum  non  antequam  evanescunt,  non  postea, 
sed    quacum    evanescunt.      Pariter    &    ratio    prima 

^  In  the  second  and  third  editions  "  pervenientis"  takes  the  place  of 
"pergentis."  In  the  third  edition  the  sentence  reads,  "  .  .  .ad 
certum  locum,  ubi  motus  finiatur,  pervenientis  velocitatem  ultimam." 

'^  In  the  second  and  third  editions  "intelligi  eam  "  takes  the  place 
of  "intelligieam." 


NE  WTON  7 

nascentium  est  ratio  quacum  nascuntur.  Et  summa 
prima  &  ultima  est  quacum  esse  (vel  augeri  et  ^ 
minui)  incipiunt  &  cessant.  Extat  limes  quem 
velocitas  in  fine  motus  attingere  potest,  non  autem 
transgredi.  Ha:c  est  velocitas  ultima  Et  par  est 
ratio  limitis  quantitatum  &  proportionum  omnium 
incipientium  &  cessantium.  Cumque  hic  limes  sit 
certus  &  definitus,  problema  est  vere  Geometricum 
eundem  determinare.  Geometrica  vero  omnia  in 
aliis  geometricis  determinandis  ac  demonstrandis 
legitime  usurpantur. 

12.  "Contendi  etiam  potest,  quod  si  dentur 
ultimse  quantitatum  evanescentium  rationes,  dabun- 
tur  &  ultimae  magnitudines  ;  &  sic  quantitas  omnis 
constabit  ex  indivisibilibus,  contra  quam  Euclides 
de  incommensurabilibus,  in  libro  decimo  Elemen- 
torum,  demonstravit.  Verum  haec  objectio  falsae 
innititur  hypothesi.  Ultimai  rationes  illae  quibuscum 
quantitates  evanescunt,  revera  non  sunt  rationes 
quantitatum  ultimarum,  sed  limites  ad  quos  quanti- 
tatum sine  limite  decrescentium  rationes  semper 
appropinquant,  &  quas  propius  assequi  possunt  quam 
prò  data  quavis  differentia,  nunquam  vero  trans- 
gredi, neque  prius  attingere  quam  quantitates 
diminuuntur  in  infinitum.  Res  clarius  intelligetur 
in  infinite  magnis.  Si  quantitates  duai  quarum  data 
est  differentia  augeantur  in  infinitum,  dabitur  harum 
ultima  ratio,  nimirum  ratio  a^qualitates  nec  tamen 
ideo  dabuntur  quantitates  ultima:^  seu  maximae 
quarum     ista    est     ratio.        Igitur     in    sequentibus, 

^   In  the  third  edition  **  aut"  takes  the  place  of  "  et," 


8  LIMITS    AND   FLUXIONS 

siquando  facili  rerum  imaginationi  ^  consulens, 
dixero  quantitates  quam  minimas,  vel  evanescentes 
vel  ultimas  ;  cave  intelligas  quantitates  magnitudine 
determinatas,  sed  cogita  semper  diminuendas  sine 
limite." 

Translation  by  Robert  Thorp  : 

13.  "Those  things  which  have  been  demonstrated 
of  curve  lines,  and  the  surfaces  which  they  compre- 
hend,  are  easily  applied  to  the  curve  surfaces  and 
contents  of  solids.  But  I  premised  these  lemmas 
to  avoid  the  tediousness  of  deducing  long  demon- 
strations  to  an  absurdity,  according  to  the  method 
of  the  ancient  geometers.  For  demonstrations  are 
rendered  more  concise  by  the  method  of  indivisibles. 
But,  because  the  hypothesis  of  indivisibles  is  some- 
what  harsh,  and  therefore  that  method  is  esteemed 
less  geometrical,  I  chose  rather  to  reduce  the 
demonstrations  of  the  following  propositions  to  the 
prime  and  ultimate  sums  and  ratios  of  nascent  and 
evanescent  quantities  ;  that  is,  to  the  limits  of  those 
sums  and  ratios  :  and  so  to  premise  the  demonstra- 
tions of  those  limits,  as  briefly  as  1  could.  For 
hereby  the  same  thing  is  performed,  as  by  the 
method  of  indivisibles  ;  and  those  principles  being 
demonstrated,  we  may  now  use  them  with  more 
safety.  Therefore,  if  hereafter  I  shall  happen  to 
consider  quantities,  as  made  up  of  particles,  or  shall 
use  little  curve  lines  for  right  ones,  I  would  not  be 
understood     to    mean     indivisible,    but    evanescent 

'   In  the  third  edition  "  conceptui-"  takes  the  place  of  "imaginationi." 


NE  WTON  9 

divisible  quantities  ;  not  the  sums  and  ratios  of 
determinate  parts,  but  always  the  limits  of  sums 
and  ratios  :  and,  that  the  force  of  such  demonstra- 
tions  always  depends  on  the  method  laid  down  in 
the  preceding  lemmas. 

14.    *'It   is   objected,    that   there   is   no   ultimate 

proportion    of  evanescent    quantities  ;    because   the 

proportion,  before  the  quantities  bave  vanished,  is 

not   ultimate  ;    and,    when   they   bave   vanished,    is 

none.      But,    by  the   same   argument,    it   might   as 

well.  be    maintained,    that    there    is    no    ultimate 

velocity  of  a  body  arriving  at  a  certain  place,  when 

its  motion  is  ended  :    because   the  velocity,    before 

the   body  arrives  at  the  place,    is  not  its  ultimate 

velocity  ;   when   it   has  arrived,    is   none.      But  the 

answer    is    easy  :    for   by   the    ultimate   velocity    is 

meant  that,  with  which  the  body  is  moved,  neither 

before  it  arrives  at  its  last  place,  when  the  motion 

ceases,  nor  after  ;  but  at  the  very  instant  when  it 

arrives  ;  that  is,  that  very  velocity  with  which  the 

body   arrives   at   its   last    place,    when    the    motion 

ceases.      And,  in  like  manner,  by  the  ultimate  ratio 

of  evanescent   quantities    is   to   be   understood    the 

ratio  of  the  quantities,  not  before  they  vanish,  nor 

after,    but   that   with   which   they   vanish.      In   like 

manner,  the  first  ratio  of  nascent  quantities  is  that 

with  which  they  begin  to  be  :  and  the  first  or  last 

sum  is  that,  with  which  they  begin  and  cease  to  be, 

or    to   be   augmented   or    diminished.      There   is   a 

limit,  which  the  veloòity  at  the  end  of  the  motion 

may    attain,     but    cannot    exceed.       This     is    the 


IO  LIMITS    AND   FLUXIONS 

ultimate  velocity.  And  there  is  a  like  lìmit  in  ali 
quantities  and  proportions  that  begin  and  cease  to 
be.  And  since  such  limits  are  certain  and  definite, 
to  determine  the  same  is  a  problem  strictly  geo- 
metrica!. But  whatever  is  geometrical  we  may  be 
allowed  to  use  in  determining  and  demonstrating 
any  other  thing  that  is  likewise  geometrical. 

15.  ''  It  may  be  also  argued,  that  if  the  ultimate 
ratios  of  evanescent  quantities  are  given,  their 
ultimate  magnitudes  will  be  also  given  ;  and  so  ali 
quantities  will  consist  of  indivisibles,  which  is.con- 
trary  to  what  Euclid  has  demonstrated  concern- 
ing  incommensurables,  in  the  tenth  book  of  bis 
Elements.  But  this  objection  is  founded  on  a  false 
supposition,  for  those  ultimate  ratios  with  which 
quantities  vanish  are  not  truly  the  ratios  of  ultimate 
quantities,  but  the  limits  to  which  the  ratios  of 
quantities,  decreasing  without  end,  always  con- 
verge ;  and  to  which  they  may  approach  nearer 
than  by  any  difference,  but  can  never  go  beyond, 
nor  attain  to,  unless  the  quantities  are  diminished 
indefinitely.  This  will  appear  more  evident  in 
quantities  indefinitely  great.  If  two  quantities, 
whose  difference  is  given,  are  augmented  continu- 
ally,  their  ultimate  ratio  will  be  given,  to  wit,  the 
ratio  of  equality  ;  but  the  ultimate  or  greatest 
quantities  themselves,  of  which  that  is  the  ratio, 
will  not  therefore  be  given.  If  then  in  what  foUows, 
for  the  more  easy  apprehension  of  things,  I  shall 
ever  mention  quantities  the  least  possihle,  or  evanes- 
cent, or  ultimate,  beware  lest  you  understand  quan- 


NE  WTON  1 1 

tities   of  a    determinate    magnitude  ;    but   conceive 
them  to  be  continually  diminished  without  limit." 

Principia,  Book  II,  Section  II,  Lemma  IL 

i6.  "  .  .  .  Has  quantitates,  ut  indeterminatas 
&  instabiles,  &  quasi  motu  fluxuve  perpetuo  cres- 
centes  vel  decrescentes,  hic  considero  ;  &  earunì  ^ 
incrementa  vel  decrementa  momentanea  sub  nomine 
momentorum  intelligo  :  ita  ut  incrementa  prò 
momentis  addititiis  seu  affirmativis,  ac  decrementa 
prò  subductitiis  seu  negativis  habeantur.  Cave 
tamen  intellexeris  particulas  finitas.  Momenta, 
quam  primum  finitae  sunt  magnitudinis,  desinunt 
esse  momenta.  Finiri  enim  repugnat  aliquatenus 
perpetuo  eorum  incremento  vel  decremento.^  In- 
telligenda  sunt  principia  jamjam  nascentia  finitarum 
magnitudinum.  Neque  enim  spectatur  in  hoc 
lemmate  magnitudo  momentorum,  sed  prima  nas- 
centium  proportio.  Eodem  recidit  si  loco  momen- 
torum usurpentur  vel  velocitates  incrementorum  ac 
decrementorum  (quas  etiam  motus,  mutationes  & 
fluxiones  quantitatum  nominare  licet)  vel  finita^ 
quajvis  quantitates  velocitatibus  bisce  proportion- 
ales.  Termini^  autem  cujusque  generantis  coeffi- 
ciens  est  quantitas,  qua^  oritur  applicando  genitam 

^  The  first  edition  gives  "  eorum  "  instead  of  "  earum," 
'•^  In  the  place  of  "  Momenta,  quam  primum  fìnitae  sunt  magnitudinis, 
desinunt  esse  momenta.  Finiri  enim  repugnat  aliquatenus  perpetuo 
eorum  incremento  vel  decremento,"  the  second  and  third  editions  have 
this  :  "  Particulae  fìnitae  non  sunt  momenta,  sed  quantitates  ipsiv  ex 
momentis  genita?." 

^  In   the  second  and  third  editions  "  Lateris  "  takes  the  place  of 
"Termini." 


12  LIMITS   AND   FLUXIONS 

ad  hunc  terminum.  Igitur  sensus  lemmatis  est, 
ut,  si  quantitatum  quarumcunque  perpertuo  motu 
crescentium  vel  decrescentium  A,  B,  C,  etc. , 
momenta,  vel  mutationum  velocitates  ^  dicantur 
a,  by  e,  etc. ,  momentum  vel  mutatio  geniti  ^  rectan- 
guli  AB  fuerit  aB-\-bA,  et  geniti  ^  contenti  ABC 
momentum  fuerit  aVtC+bAC^cKB  :  .    .    . 

ly.  '^  Cas.  I.  Rectangulum  quodvis  motu  perpetuo 
auctum  AB,  ubi  de  lateribus  A  &  B  deerant  momen- 
torum  dimidia*  ^a  &  ^b,  fuit  A  —  ^a  in  B  —  ^b, 
seu  AB  —  ^aB  —  h  bA  +  \ab  \  &  quam  primum  latera 
A  &  B  alteris  momentorum  dimidiis  aucta  sunt, 
evadit  A  +  |  ^  in  B  +  |-  b,  seu  AB  +  |  ^B  +  J  ^ A  +  ^  ab. 
Uè  hoc  rectangulo  subducatur  rectangulum  prius,  et 
manebit  excessus  ^B  +  ^A.  Igitur  laterum  incre- 
mentis  totis  ^  et  ^  generatur  rectanguli  incrementum 
aB-^bA.      Q.E.D." 

English  Translation  by  Andrew  Motte  : 

i8.  "  .  .  .  These  quantities  I  bere  consider  as 
variable  and  indetermined  and  increasing  or  decreas- 
ing  as  it  were  by  perpetuai  motion  or  flux  ;  and  I 
understand  their  momentaneous  increments  or 
decrements  by  the  name  of  Moments  ;  so  that  the 

^  In  the  second  and  third  editions  "hoc  latus  "  takes  the  place  of 
"hunc  terminum." 

"  In  the  third  edition  "  vel  his  proportionales  mutationum  veloci- 
tates" takes  the  places  of  "vel  mutationum  velocitates." 

'  "geniti"  is  left  out  in  the  first  edition. 

^  In  this  history,  the  solidus  (  /  )  will  be  used  sometimes  in  printing 
fractions  which  come  in  the  line  of  the  text.  The  reader  must  remem- 
ber  that  this  notation  is  modem  ;  it  occurs  in  none  of  the  passaores 
which  we  quote  from  seventeenth-  and  eighteenth-century  books.  In 
some  cases  the  use  of  the  solidus  has  made  it  necessary  to  insert  paren- 
theses  which  do  not  occur  in  the  originai. 


NEWTON  13 

increments  may  be  esteemed  as  added,  or  affirm- 
ative  moments  ;  and  the  decrements  as  subducted, 
or  negative  ones.  But  take  care  not  to  look  upon 
finite  particles  as  such.  Finite  particles  are  not 
moments,  but  the  very  quantities  generated  by  the 
moments.  We  are  to  conceive  them  as  the  just 
nascent  principles  of  finite  magnitudes.  Nor  do  we 
in  this  Lemma  regard  the  magnityde  of  the  moments, 
but  their  first  proportion  as  nascent.  It  will  be  the 
same  thing,  if,  instead  of  moments,  we  use  either 
the  Velocities  of  the  increments  and  decrements 
(which  may  also  be  called  the  motions,  mutations, 
and  fluxions  of  quantities)  or  any  finite  quantities 
proportional  to  those  velocities.  The  coefficient  of 
any  generating  side  is  the  quanti ty  which  arises  by 
applying  the  Genitum  to  that  side.  Wherefore  the 
sense  of  the  Lemma  is,  that  if  the  moments  of  any 
quantities  A,  B,  C,  etc,  increasing  or  decreasing  by 
a  perpetuai  flux,  or  the  velocities  of  the  mutations 
which  are  proportional  to  them,  be  called  a,  b,  e,  etc, 
the  moment  or  mutation  of  the  generated  rectangle 
AB  will  be  ^B  +  ^A  ;  the  moment  of  the  generated 
content  ABC  will  be  ^BC  +  ^AC  +  ^AB  :  .    .    . 

19.  ''  Case  I.  Any  rectangle  as  AB  augmented  by 
a  perpetuai  flux,  when,  as  yet,  there  wanted  of  the 
sides  A  and  B  half  their  moments  \a  and  \b,  was 
A  -  i  «  into  Vi-\b,  ox  K^-\a^-\bh^-\ab\  but 
as  soon  as  the  sides  A  and  B  are  augmented  by  the 
other  half  moments  ;  the  rectangle  becomes  t\.-\-\a 
into  B  +  J  ^,  or  AB  +  è  ^B  +  i  ^ A  +  1  ab.  From  this 
rectangle   subduct   the   former  rectangle,  and  there 


14  LIMITS    AND   FLUXIONS 

wìU  remain  the  excess  aY>  +  /^A.  Therefore  with  the 
whole  increments  a  and  b  of  the  sides,  the  increment 
aV>  +  ^A  of  the  rectangle  is  generated.      Q.  E.  D.  " 

IL   Wallis's  De  Algebra  Tractatus 

20.  The  Latin  edition  of  John  Wallis's  Algebra, 
which  appeared  in  1693,  contains  on  pages  390-396 
a  treatise  on  the  "Quadrature  of  Curves  "  which 
Newton  had  prepared  many  years  before,  and  from 
which  he  cited  many  things  in  his  letter  of  October 
24,  1676.  In  revised  phraseology  and  with  a  new 
Introduction,  the  "Quadrature  of  Curves"  was 
republished  in  1704,  as  we  shall  see  presently. 
Through  the  researches  of  Rigaud  ^  we  know  now 
that  what  is  given  in  Wallis's  Algebra,  p.  390,  line 
18,  to  p.  396,  line  19,  are  Newton's  own  words, 
except,  no  doubt,  the  word  "  clarissimus,"  as 
applied  to  himself  From  this  part  we  quote  as 
foUows  :  2 — 

21.  Page  391:  ''V^x  fluentes  qua^ititates  intelli- 
git  indeterminatas,  id  est  quae  in  generatione 
Cuvarum  per  motum  localem  perpetuo  augentur  vel 
diminuuntur,  &  per  earum  fluxionem  intelligit  celeri- 
tatem  incrementi  vel  decrementi.  Nam  quamvis 
fluentes  quantitates  &  earum  fluxiones  prima  fronte 
conceptu  difficiles  videantur,  (solent  enim  nova 
difficilius  concipi),  earundem  tamen  notionem  cito 
faciliorem  evasuram  putat,  quam  sit  notio  momen- 

'  S.  P.  Rigaud,  Historical  Essay  on  Sir  Isaac  Newton^ s  Principia^ 
Oxford,  1838,  p.  22. 

2  Johannis  Wallis,  S.T.D.,  De  Algebra  Tractatus;  Historicus  &f 
Practicus.     Oxoniae,  MDCXCIII. 


NEWTON  15 

torum  aut  paì'tium  ininimarum  vel  differentiaruin 
infinite par-varum  :  propterea  quod  figurarum  &  quan- 
titatum  generatio  per  motum  continuum  magis 
naturalis  est  &  facilius  concipitur,  &  Schemata  in 
hac  methodo  solent  esse  simpliciora,  quam  in  illa 
partium.    ..." 

22.  Page  392:  "  Sint  v,  x,  y,  ^  fluentes  quanti- 
tates,  &  earum  fluxiones  his  notis  i),  x,  j>,  i  desig- 
nabuntur  respective.  Et  quoniam  hae  fluxiones  sunt 
etiam  indeterminata^  quantitates,  &  perpetua  muta- 
tione  redduntur  majores  vel  minores,  considerat 
velocitates  quibus  augentur  vel  diminuuntur  tanquam 
earum  fluxiones,  &  punctis  binis  notat  in  hunc 
modum  i»,  ir,  j,  z,  &  perpetuum  incrementum  vel 
decrementum  harum  fluxionum  considerat  ut  ipsarum 
fluxiones,    ..." 

23.  Page  392:  "  Sit  enim  0  quantitas  infinite 
parva,  &  sint  02,  oj,  ox  Synchrona  momenta  seu 
incrementa  momentanea  quantitatum  fluentium  z,  y, 
8i  X  :  &  hae  quantitates  proximo  temporis  momento 
per  accessum  incrementorum  momentaneorum  eva- 
dent  z-\-oz,  y-\-oy,  x-{-ox\  ..."  After  substitut- 
ing  these  in  x'^  —  xyy-^aaz  =  o,  then  subtracting  the 
originai  expression  and  dividing  the  remainder  by 
0,  he  remarks  (page  393)  :  ''  Terminos  multiplicatos 
per  0  tanquam  infinite  parvos  dele,  &  manebit 
aequat i o   3i'.r2  _  ^^  _  2xyy  +  aaè  =  o.  " 

Translation  : 

24.  Page  391  :  "  By  flowing  quantities  he  under- 
stands  indeterminates,  that  is,  those  which,  in  the 


i6  LIMITS   AND   FLUXIONS 

generation  of  curves  by  locai  motion  are  always 
increased  or  diminished,  and  by  their  fluxions  he 
understands  the  velocity  of  increase  or  decrease. 
For,  however  difficult  of  comprehension  flowing 
quaìttities  and  their  fluxions  appear  at  first  sight 
(for  new  things  are  usually  perceived  with  diffi- 
culty),  yet  he  thinks  a  notion  of  them  will  be 
obtained  more  easily  than  the  notion  of  moments 
either  oi  least  pai'ts  or  of  infinitely  small  diffei^ences  ; 
because  the  generation  of  figures  and  quantities  is 
more  naturally  and  easily  conceived,  and  the  draw- 
ings  in  this  method  are  usually  more  simple  than  in 
that  of  parts." 

25.  Page  392:  "  Let  the  flowing  quantities  be 
designated  v,  x^  j',  2,  and  their  fluxions  by  the 
marks  v,  x,  j>,  i,  respectively.  And  since  these 
fluxions  are  likewise  indeterminate  quantities,  and 
by  perpetuai  motion  become  greater  or  lesser,  he 
considers  the  velocities  by  which  they  are  increased 
or  diminished  as  their  fluxions,  and  marks  them 
with  doublé  dots  in  this  way  i),  x,  y,  i,  and  he  con- 
siders the  perpetuai  increase  or  decrease  of  these 
fluxions  as  fluxions  of  themselves.    ..." 

26.  Page  392:  **  Let  0  be  an  infinitely  small 
quantity,  and  02^  oy,  ox  the  synchronous  moments 
or  momentaneous  increments  of  the  flowing  quanti- 
ties z,  y,  X  \  and  these  quantities  at  the  next 
moment  of  time,  by  the  accession  of  the  momen- 
taneous increments  become  z-\-02,  y-\-oy,  x  +  ox: 
.  .  ."  After  substituting  these  in -r3—;i7j/-j-^(3:<3'  =  o, 
then  subtracting  the  originai  expression  and  divid- 


NEWTON  17 

ing  the  remainder  by  0,  he  remarks  (page  393)  : 
''Destroy  the  terms  multipUed  by  0  as  infinitely 
small,  and  there  will  remain  the  equation  2)^x^—^yy 
—  2X}'y  +  aaà  =  o.  " 

III.  Quadratura  Curvarum,^  1704 
''  Introductio 

27.  "  Quantitates  Mathematicas  non  ut  ex  parti- 
bus  quam  minimis  constantes,  sed  ut  motu  con- 
tinuo descriptas  hic  considero.  Lineae  describuntur 
ac  describendo  generantur  non  per  appositionem 
partium  sed  per  motum  continuum  punctorum, 
superficies  per  motum  Hnearum,  soHda  per  motum 
superficierum,  anguli  per  rotationem  laterum,  tem- 
pora per  fluxum  continuum,  et  sic  in  ca^teris.  Hse 
Geneses  in  rerum  natura  locum  vere  habent  et  in 
motu  corporum  quotidie  cernuntur.  Et  ad  hunc 
modum  Veteres  ducendo  rectas  mobiles  in  longi- 
tudinem  rectarum  immobilium  genesin  docuerunt 
rectangulorum. 

28.  "Considerando  igitur  quod  quantitates  aequa- 
libus  temporibus  crescentes  et  crescendo  genita^, 
prò   velocitate   majori   vel    minori   qua   crescunt   ac 

^  Tractatus  de  Quadratura  Curvarum,  published  in  1704  in  London, 
as  an  appendix  to  Newton's  Opiicks.  It  was  reprinted  under  the 
editorship  of  William  Jones  in  London  in  ihe  year  171 1,  in  a  volume 
containing  also  three  other  papers  of  Newton,  viz.,  the  De  analysi per 
ceguationes  infinifas,  Enumcratio  Hnearum  tertii  ordinis,  and  Methodus 
differentialis.  An  English  translation  of  the  Qìiadratura  Curvarum, 
made  by  John  Stewart,  was  brought  out  in  1745  at  London,  in  a  volume 
containing  also  Newton's  Analysis  hy  Equations  of  an  Infinite  Ntiinber 
of  Ternis.  A  German  translation  of  the  Quadratura  Curvarum  by 
Gerhard  Kowalewski  appeared  at  Leipzig  in  1908  in  OsiwalcP s  Klassiker 
der  exakten  Wissenschaften,  Nr.  164. 

2 


i8 


LIMITS    AND   FLUXIONS 


generantur,  evadunt  majores  vel  minores  ;  metho- 
dum  quaerebam  determinandi  quantitates  ex  veloci- 
tatibus  motuum  vel  incrementorum  quibus  gener- 
antur; et  has  motuum  vel  incrementorum  velocitates 
nominando  Fluxiones  et  quantitates  genitas  nomin- 
ando FluenteSy  incidi  paulatim  Annis  1665  et  1666 
in  Methodum  Fluxionum  qua  hic  usus  sum  in 
Quadratura  Curvarum. 

29.    ''  Fluxiones  sunt  quam  proxime  ut  Fluentìum 


^ 

f 

rc__.£ 

V           A 

B       'A 

: 

( 

FiG.  : 

5        a 

2. 

augmenta  aequalibus  temporis  particulis  quam  mini- 
mis  genita,  et  ut  accurate  loquar,  sunt  in  prima 
ratione  augmentorum  nascentium  ;  exponi  autem 
possunt  per  lineas  quascunque  quae  sunt  ipsis  pro- 
portionales.  Ut  si  areae  ABC,  ABDG  ordinatis 
BC,  BD  super  basi  AB  uniformi  cum  motu  pro- 
gredientibus  describantur,  harum  arearum  fluxiones 
erunt  inter  se  ut  ordinata^  describentes  BC  et  BD, 
et  per  ordinatas  illas  exponi  possunt,  propterea  quod 
ordinatae   illae  sunt  ut  arearum  augmenta  nascentia. 


NEWTON  19 

Progrediatur  ordinata  BC  de  loco  suo  BC  in  locum 
quemvis  novum  bc.  Compleatur  parallelogrammum 
BCE<^,  ac  ducatur  recta  VTH  qua^  curvam  tangat 
in  C  ibsisque  bc  et  BA  productis  occurrat  in  T  et 
V  :  et  abscissai  AB,  ordinata^  BC,  et  lineae  curvae 
AC^  augmenta  modo  genita  erunt  B^,  E^,  et  Qc  ;  et 
in  horum  augmentorum  nascentium  ratione  prima 
sunt  latera  trianguli  CET,  ideoque  fluxiones  ipsarum 
AB,  BC  et  AC  sunt  ut  trianguli  illius  CET  latera 
CE,  ET  et  CT  et  per  eadem  latera  exponi  possunt, 
vel  quod  perinde  est  per  latera  trianguli  consimilis 
VBC. 

30.  ''Eodem  recidit  si  sumantur  fluxiones  in 
ultima  ratione  partium  evanescentium.  Agatur 
recta  Qc  et  producatur  eadem  ad  K.  Redeat  ordinata 
bc  in  locum  suum  priorem  BC,  et  cceuntibus  punctis 
C  et  e,  recta  CK  coincidet  cum  tangente  CH,  et 
triangulum  evanescens  CE^  in  ultima  sua  forma 
evadet  simile  triangulo  CET,  et  ejus  latera  evanes- 
centia  CE,  E<;  et  C^  erunt  ultimo  inter  se  ut  sunt 
trianguli  alterius  CET  latera  CE,  ET  et  CT,  et 
propterea  in  hac  ratione  sunt  fluxiones  linearum 
AB,  BC  et  AC.  Si  puncta  C  et  e  parvo  quovis 
intervallo  ab  invicem  distant  recta  CK  parvo  inter- 
vallo a  tangente  CH  distabit.  Ut  recta  CK  cum 
tangente  CH  coincidat  et  rationes  ultimai  linearum 
CE,  Y.C  et  C^  inveniantur,  debent  puncta  C  et  ^ 
coire  et  omnino  coincidere.  Errores  quam  minimi 
in  rebus  mathematicis  non  sunt  contemnendi. 

31.  "  Simili  argumento  si  circulus  centro  B  radio 
BC   descriptus    in    longitudinem    abscissae    AB    ad 


20  LIMITS    AND   FLUXIONS 

angulos  rectos  uniformi  cum  motu  ducatur,  fluxio 
solidi  geniti  ABC  erit  ut  circulus  ille  generans,  et 
fluxio  superficiei  ejus  erit  ut  perimeter  circuii  illius 
et  fluxio  lineae  curvae  AC  conjunctim.  Nam  quo 
tempore  solidum  ABC  generatur  ducendo  circulum 
illum  in  longitudinem  abscissa:^  AB,  eodem  super- 
ficies  ejus  generatur  ducendo  perimetrum  circuii  illius 
in  longitudinem  curvai  AC.    .    .    . 

32.  ''  Fluat  quantitas  X  uniforiniier  et  inv emenda 
sit  fluxio  quafititatis  x'\  Quo  tempore  quantitas  x 
fluendo  evadit  x-\-o,  quantitas  x'"-  evadet  x-\-o\**,  id 
est  per  methodum  serierum  infinitarum,  x"-\-nox"~'^ 
-\-{nn  —  n)l  2  oox''~^ -\-etc.  Et  augmenta  0  et  nox'"-'^ 
-\-{nn  —  n)  /  2  <9<?,r''~^  +  etc.sunt  ad  invicem  ut  i  et  nx''~^ 
■\-{nn  —  n)\2  ox"-^-\-etc.  Evanescant  jam  augmenta 
illa,  et  eorum  ratio  ultima  erit  i  ad  nx"~'^  :  ideoque 
fluxio  quantitatis  x  est  ad  fluxionem  quantitatis  x'^ 
ut  I  ad  nx"~^. 

33.  "Similibusargumentis  per  methodum  rationum 
primarum  et  ultimarum  colligi  possunt  fluxiones 
linearum  seu  rectarum  seu  curvarum  in  casibus 
quibuscunque,  ut  et  fluxiones  superficierum,  angu- 
lorum  et  aliarum  quantitatum.  In  finitis  autem 
quantitatibus  Analysin  sic  instituere,  et  finitarum 
nascentium  vel  evanescentium  rationes  primas 
vel  ultimas  investigare,  consonum  est  geometriae 
veterum  :  et  volui  ostendere  quod  in  Methodo 
Fluxionum  non  opus  sit  figuras  infinite  parvas 
in  geometriam  introducere.  Peragi  tamen  potest 
Analysis  in  figuris  quibuscunque  seu  finitis  seu 
infinite  parvis  quac  figuris  cvanescentibus  finguntur 


NEWTON  21 

similes,    ut    et    in    figuris   quae    prò    infinite    parvis 
haberi  solent,  modo  caute  procedas." 

Translation  by  John  Stewart  : 

*  '  Intì'oduction 

34.  *'I  consider  mathematica!  quantities  in  this 
place  not  as  consisting  of  very  small  parts  ;  but 
as  describ'd  by  a  continued  motion.  Lines  are 
describ'd,  and  thereby  generated  not  by  the  appo- 
sition  of  parts,  but  by  the  continued  motion  of 
points  ;  superficies's  by  the  motion  of  lines  ;  soHds 
by  the  motion  of  superficies's  ;  angles  by  the  rota- 
tion  of  the  sides  ;  portions  of  time  by  a  continuai 
flux  :  and  so  in  other  quantities.  These  geneses 
really  take  place  in  the  nature  of  things,  and  are 
daily  seen  in  the  motion  of  bodies.  And  after  this 
manner  the  ancients,  by  drawing  moveable  right 
lines  along  immoveable  right  lines,  taught  the 
genesis  of  rectangles. 

35.  **  Therefore  considering  that  quantities,  which 
increase  in  equal  times,  and  by  increasing  are 
generated,  become  greater  or  less  according  to  the 
greater  or  less  velocity  with  which  they  increase 
and  are  generated  ;  I  sought  a  method  of  determin- 
ing  quantities  from  the  velocities  of  the  motions 
or  increments,  with  which  they  are  generated  ;  and 
calling  these  velocities  of  the  motions  or  increments 
Fluxions,  and  the  generated  quantities  Fluents^  I 
fell  by  degrees  upon  the  Method  of  Fluxions,  which 
I  have  made  use  of  here  in  the  Quadrature  of 
Curves,  in  the  years  1665  and  1666. 


22  LIMITS    AND  FLUXIONS 

36.  ''Fluxions  are  very  nearly  as  the  augments 
of  the  fluents  generateci  in  equal  but  very  small 
particles  of  time,  and,  to  speak  accurately,  they  are 
in  the  first  ratio  of  the  nascent  augments  ;  but  they 
may  be  expounded  by  any  lines  which  are  pro- 
portional  to  them. 

37.  ''Thus  if  the  area's  ABC,  ABDG  be 
described  by  the  ordinates  BC,  BD  moving  along 
the  base  AB  with  an  uniform  motion,  the  fluxions  of 
these  area's  shall  be  to  one  another  as  the  describ- 
ing  ordinates  BC  and  BD,  and  may  be  expounded  by 
these  ordinates,  because  that  these  ordinates  are  as 
the  nascent  augments  of  the  area's. 

38.  "  Let  the  ordinate  BC  advance  from  it's 
place  into  any  new  place  bc.  Complete  the  par- 
allelogram  BCE^,  and  draw  the  right  line  VTH 
touching  the  curve  in  C,  and  meeting  the  two  lines 
bc  and  BA  produc'd  in  T  and  V  :  and  B^,  E<:  and 
Qc  will  be  the  augments  now  generated  of  the 
absciss  AB,  the  ordinate  BC  and  the  curve  line 
AC^;  and  the  sides  of  the  triangle  CET  are  in  the 

first  ratio  of  these  augments  considered  as  nascent, 
therefore  the  fluxions  of  AB,  BC  and  AC  are  as 
the  sides  CE,  ET  and  CT  of  that  triangle  CET, 
and  may  be  expounded  by  these  same  sides,  or, 
which  is  the  same  thing,  by  the  sides  of  the  triangle 
VBC,  which  is  similar  to  the  triangle  CET. 

39.  '*lt  Comes  to  the  same  purpose  to  take  the 
fluxions  in  the  ultimate  ratio  of  the  evanescent 
parts.  Draw  the  right  line  Qc,  and  produce  it  to 
K.      Let    the    ordinate    bc  return    into    it's   former 


NEWTON  23 

place  BC,  and  when  the  points  C  and  e  coalesce, 
the  right  line  CK  will  coincide  with  the  tangent 
CH,  and  the  evanescent  triangle  CE<:in  it's  ultimate 
form  will  become  similar  to  the  triangle  CET,  and 
it's  evanescent  sides  CE,  E^  and  Qc  will  be  ulti- 
mately  among  themselves  as  the  sides  CE,  ET  and 
CT  of  the  other  triangle  CET,  are,  and  therefore 
the  fluxions  of  the  lines  AB,  BC  and  AC  are  in  the 
same  ratio.  If  the  points  C  and  e  are  distant  from 
one  another  by  any  small  distance,  the  right  line 
CK  will  likewise  be  distant  from  the  tangent  CH 
by  a  small  distance.  That  the  right  line  CK  may 
coincide  with  the  tangent  CH,  and  the  ultimate 
ratios  of  the  lines  CE,  E^  and  Qc  may  be  found,  the 
points  C  and  e  ought  to  coalesce  and  exactly  co- 
incide. The  very  smallest  errors  in  mathematical 
matters  are  not  to  be  neglected. 

40.  ''By  the  like  way  of  reasoning,  if  a  circle 
describ'd  with  the  center  B  and  radius  BC  be  drawn 
at  right  angles  along  the  absciss  AB,  with  an  uni- 
form  motion,  the  fluxion  of  the  generated  solid 
ABC  will  be  as  that  generating  circle,  and  the 
fluxion  of  it's  superficies  will  be  as  the  perimeter  of 
that  circle  and  the  fluxion  of  the  curve  line  AC 
jointly.  Por  in  whatever  time  the  solid  ABC  is 
generated  by  drawing  that  circle  along  the  length 
of  the  absciss,  in  the  same  time  it's  superficies  is 
generated  by  drawing  the  perimeter  of  that  circle 
along  the  length  of  the  curve  AC.    ..." 

41.  '''  Let  the  quantity  x  flow  uniformly,  and  let  it 
b  e  propose  ci  to  find  the  fluxion  of  x". 


24  LIMITS    AND   FLUXIONS 

''In  the  same  time  that  the  quantity  x,  by 
flowing,  becomes  x-\-o^  the  quantity  x"^  will 
become  x-^-o^,  that  is,  by  the  method  of  infinite 
series's,  x''-^nox''-^-\-{n^ —  n)l2  oox''-^-\-etc.  And 
the  augments  6>  and  nox*'-^-{-{n'^ —  n)l2  oox''-^-{-etc. 
are  to  one  another  as  i  and  nx''-'^-j-(n^ —  n)l2  ox^-'^ 
+  etc.  Now  let  these  augments  vanish,  and  their 
ultimate  ratio  will  be  i  to  nx""-^. 

42.  "  By  like  ways  of  reasoning,  the  fluxions  of 
lines,  whether  right  or  curve  in  ali  cases,  as  likewise 
the  fluxions  of  superficies's  angles  and  other  quan- 
tities,  may  be  collected  by  the  method  of  prime  and 
ultimate  ratios.  Now  to  institute  an  analysis  after 
this  manner  in  finite  quantities  and  investigate  the 
prime  or  ulti^nate  ratios  of  these  finite  quantities 
when  in  their  nascent  or  evanescent  state,  is  con- 
sonant  to  the  geometry  of  the  ancients  :  and  I  was 
willing  to  show  that,  in  the  Method  of  Fluxions, 
there  is  no  necessity  of  introducing  figures  infinitely 
small  into  geometry.  Yet  the  analysis  may  be 
performed  in  any  kind  of  figures,  whether  finite  or 
infinitely  small,  which  are  imagin'd  similar  to  the 
evanescent  figures  ;  as  likewise  in  these  figures, 
which,  by  the  Method  of  Indivisibles,  used  to  be 
reckoned  as  infinitely  small,  provided  you  proceed 
with  due  caution." 

43.  In  the  Quadrature  of  Curves  proper,  under 
'*  Proposition  I"  the  proof  of  the  rule  for  finding 
the  fluxion  of  expressions  like  x^  —  Xf^ -^  a'^z  —  l?^  =  o 
contains  the  following  passages  which  indicate  the 
use    made    of  the    symbol    "  (?  "    and    of  the    term 


NEWTON  25 

'*  moment,"  and  the  mode  of  passing  to  the  limit. 
We  quote  : — 

*  *  Demonstratio 

44.  **Nam  sit  o  quantitas  admodum  parva  et 
sunto  oz,  oy,  ox,  quantitatum  z,  y,  x,  momenta  id 
est  incrementa  momentanea  synchrona.  Et  si 
quantitates  fluentes  jam  sunt  z,  y  et  x,  hae  post 
momentum  temporis  incrementis  suis  oz,  oy,  ox 
auctae,  evadent  z-\-oz^  y-\-oy,  x-\-ox,  quae  in 
aequatione  prima  prò  z^  y  Qt  x  scriptae  dant 
aequationem  ... 

'^xx'^  +  ^xxox  -\-  i'^oo  —  xyy  —  2xyy 

—  2xoyy  —  xoyy  —  xooyy  +  aaz  =  o. 

Minuatur  quantitas  0  in  infinitum,  et  neglectis 
terminis  evanescentibus  restabit  ^xx"^  —  xyy  —  2xyy 
-^aaò=:0.     O.E.U." 

Translation  by  John  Stewart  : 

''  Demonstration 

45.  "  For  let  6»  be  a  very  small  quantity,  and  let 
oz,  oy,  oxhe  the  moments,  that  is  the  momentaneous 
synchronal  increments  of  the  quantities  z,  y,  x. 
And  if  the  flowing  quantities  are  just  now  z,  y,  x, 
then  after  a  moment  of  time,  being  increased  by 
their  increments  oz,  oy,  ox  :  these  quantities  shall 
become  z-\-oz,  y-\-oy,  x-\-ox:  which  being  wrote 
in  the  first  equation  for  z,  y  and  x,  give  this 
equation  .  .  . 

^xx^  +  ^xxox  +  ,i^oo  —  xyy  —  2xyy 

—  2xoyy  —  xoyy  —  xooyy  +  aaz  —  o. 


26  LIMITS   AND   FLUXIONS 

Let  the  quantity  o  be  diminished  infinitely,  and 
neglecting  the  terms  which  vanish,  there  will  remain 
^xx^  —  xyy  —  2xyy  +  aaz  =  o.      Q.  E.  D.  " 

IV.  An  Account  of  the  '*Commercium 
Epistolicum  " 

46.  It  is  now  generally  accepted  that  the  account  ^ 
of  the  Commeixium  Epistolicum^  published  in  the 
Philosophical  Transactions,  London,  17 17,  was 
written  by  Newton.  The  reasons  for  attributing  it 
to  him  are  stated  by  De  Morgan^  and  by  Brewster.^ 
In  abstract  the  account  is  as  follows  : — 

47.  (Pp.  177-178.)  In  a  letter  of  October  24, 
1676,  to  Oldenburgh,  Newton  explained  that  in 
deducing  areas  he  considered  the  area  as  growing 
"by  continuai  Flux";  ''from  the  Moments  of  Time 
he  gave  the  Name  of  Moments  to  the  momentaneous 
Increases,  or  infinitely  small  Parts  of  the  Abscissa 
and  Area  generated  in  Moments  of  Time.  The 
Moment  of  a  Line  he  called  a  Point,  in  the  Sense 
of  Cavalerius,  tho'  it  be  not  a  geometrical  Point, 
but  a  Line  infinitely  short,  and  the  Moment  of  an 
Area  or  Superficies  he  called  a  Line,  in  the  sense 
of  Cavalerius,    tho'   it   be   not  a  geometrical   Line, 

^  Philosophical  Transactions,  voi.  xxix,  for  the  years  17 14,  17 15, 
1716.  Li)ndon,  1717.  "  An  Account  of  the  Hook  entituled  Commer- 
ciimi  Epistolicum  Collimi  et  aliorum,  De  Analysi  promoia  .  .  .," 
pp.  173-224.  This  account  was  translated  into  Latin  and  inserted  in 
the  edilion  of  the  Commerciiim  Epistolicum  of  1725. 

2  See  De  Morgan's  articles  in  the  Philosophical  Magazine,  S.  4, 
voi.  iii,  June,   1852,  pp.  440-444;  v  )1.  iv,   November   1652,   p.   323. 

^  Sir  David  V>tcvi^\.ttx,  Memoirs  of  the  Life,  Writings,and  Discoveries 
of  Sir  Isaac  Newton,  2nd  ed.,  voi.  ii,  Edinburgh,  1860,  pp.  35,  36. 


NE  WTON  27 

but  a  Superficies  infinitely  narrow.  And  vvhen  he 
consider'd  the  Ordinate  as  the  Moment  of  the  Area, 
he  understood  by  it  the  Rectangles  under  the  geo- 
metrica! Ordinate  and  a  Moment  of  the  Abscissa, 
tho'  that  Moment  be  not  always  expressed."  Again, 
p.  179:  "And  this  is  the  Foundation  of  the 
Method  of  Fluxions  and  Moments,  which  Mr. 
Newton  in  his  Letter  dated  Octob.  24,  1676, 
comprehended  in  this  Sentence.  Data  cequatione 
quotcunque  fluentes  quantitates  involvente,  invenire 
Fluxiones  ;  et  vice  versa.  In  this  Compendium 
Mr,  Newton  represents  the  uniform  Fluxion  of 
Time,  or  of  any  Exponent  of  Time  by  an  Unit  ; 
the  Moment  of  Time  or  its  Exponent  by  the  Letter 
o  ;  the  Fluxions  of  other  Quantities  by  any  other 
Symbols  ;  the  Moments  of  those  Quantities  by  the 
Rectangles  under  those  Symbols  and  the  Letter  0  ; 
and  the  Area  of  the  Curve  by  the  Ordinate  inclosed 
in  a  Square,  the  Area  being  put  for  a  Fluent  and 
the  Ordinate  for  its  Fluxion.  When  he  is  demon- 
strating  a  Proposition  he  uses  the  Letter  0  for  a 
finite  Moment  of  Time,  or  of  its  Exponent,  or  of 
any  Quantity  flowing  uniformly,  and  performs  the 
whole  Calculation  by  the  Geometry  of  the  Ancients 
in  finite  Figures  or  Schemes  without  any  Approxi- 
mation  :  and  so  soon  as  the  Calculation  is  at  an 
End,  and  the  Equation  is  reduced,  he  supposes  that 
the  moment  0  decreases  in  infiìiitum  and  vanishes. 
But  when  he  is  not  demonstrating  but  only  investi- 
gating  a  Proposition,  for  making  Dispatch  he 
supposes  the   Moment  0  to  be  infinitely  little,  and 


28  LIMirS    AND   FLUXIONS 

forbears   to  write  it  down,  and  uses  ali  manner  of 
Approximations   which    he   conceives   will    produce 
no  Error  in  the  Conclusion.  "    In  ^^yN\.ovC^  Principia 
"  he  frequently  considers  Lines  as  Fluents  described 
by   Points,    whose  Velocities   increase  or   decrease, 
the    Velocities    are    the    first    Fluxions,    and    their 
Increase    the    second.  "      The   Compendium   of  his 
Analysis  was  written  ''in  or  before  the  year  1669" 
(p.   180).      ''And  the  same  Way  of  working  he  used 
in  his   Book  of  Quadratures,  and  stili  uses  to  this 
day"(p.   182).      On  p.  204  we  read  :    '*Mr.  Newton 
used  the  letter  0  in  his  Analysis  written  in  or  before 
the  Years   1669,  and  in   his   Book  of   Quadratures, 
and  in  his  Principia  PhilosophicB,   and  stili  uses  it 
in   the   very   same    Sense   as    at    first.    .    .    .    These 
Symbols  0  and  i' are  put  for  things  of  a  different  kind. 
The    one    is    a    Moment,    the    other    a   Fluxion   or 
Velocity  as  has  been  explained  above.    .    .    .     Prickt 
Letters  never  signify  Moments,    unless  when  they 
are  multiplied  by  the  Moment  0  either  exprest  or 
understood  to  make  them  infinitely  little,  and  then 
the  Rectangles  are  put  for  the  Moments  "  (p.  204). 
Further  on  we  read  :    "  It  [the  method  of  fluxions] 
is   more   elegant  [than   the   Differential    Method   of 
Leibniz],  because  in  his  Calculus  there  is  but  one 
infinitely  little  Quantity  represented  by  a  symbol, 
the  symbol  0.      We  bave  no  Ideas  of  infinitely  little 
Ouantities,   and   therefore   Mr.    Newton   introduced 
Fluxions  into  his  Method,  that  it  might  proceed  by 
finite  Ouantities 'as  much  as  possible.      It  is  more 
Naturai  and  Geometrical,  because  founded  upon  the 


NE  WTON  29 

primce  quantitatum  nascentiuin  rationes,  which  have 
a  Being  in  Geometry,  whilst  Indivisibles,  upon 
which  the  Differential  Method  is  founded,  have  no 
Being  either  in  Geometry  or  in  Nature.  There  are 
rationes  primce  quantitatum  nascentium,  but  not 
quantitates  prinicB  nascentes.  Nature  generates 
Quantities  by  continuai  Flux  or  Increase  ;  and  the 
ancient  Geometers  admitted  such  a  Generation  of 
Areas  and  Solids  "  (p.    205). 

Front  Newton' s  Correspondence  and  Manuscripts 
not  in  print  in  1734 

48.  Manuscripts  of  Newton,  some  of  them  stili 
unpublished,  show  that  he  first  thought  of  fluents 
and  fluxions  in  1665  and  1666,  when  he  was  in 
his  twenty-third  and  twenty-fourth  years.^  The 
notation  by  dots  occurs  as  early  as  1665.  As 
pointed  out  by  De  Morgan,-  these  early  papers  are 
infinitesimal  in  character.  They  were  first  published 
in  1838.^  A  manuscript,  dated  Nov.  13,  1665, 
gives  rules  for  finding  the  velocities  p,  q,  ;-,  etc. , 
of  two  or  more  lines  x,  y,  z,  etc,  described  by  bodies 
A,  B,  C,  etc,  the  lines  being  related  to  each  other 

^  See  a  list  of  Newton's  manuscripts  and  publications  on  fiuxional 
calculus  prepared  by  Philip  E.  B.  Jouidain,  in  his  edition  of  Augtistus 
De  Morgati's  Essays  on  the  Life  and  Work  of  Nezvton,  The  Open 
Court  Publishing  Co.,  1914,  pp.  107- 1 12. 

"  Augustus  De  Morgan,  "On  the  Early  History  of  Infinitesimals 
in  England,"  7'he  London,  Edinburgh,  and  Duhlin  Philosophical 
Magazine,  4th  S.,  voi.  iv,  1852,  pp.  321-330.  This  article  is  an 
important  historical  contribution,  of  which  extensive  use  is  made  in  the 
present  history. 

^  See  S.  P.  Rigaud,  Historical  Essay  on  the  first  Public ation  of  Sir 
Isaac  Newton's  Principia,  Oxford,  1838,  Appendix,  pp.  20-24. 


30  LIMITS    AND   FLUXIONS 

by  an  equation,  such  ^i?,  x^—2a^y-\- zzx—yyx-\-zyy  — 
z'^—o.  ''If  the  body  A,  with  the  velocity  /, 
describe. the  infinitely  Httle  line  o  in  one  moment, 
in  the  same  moment  B,  with  the  velocity  q,  will 
describe  the  line  oq  /  /,"  and  the  body  C,  with  the 
velocity  r,  will  describe  the  line  or  /  /.  So  that, 
if  the  described  lines  be  x,  y  and  z  "in  one 
moment,"  they  will  be  x-\-o,  y-\-oq  / /,  z  +  or  j  p 
'*in  the  next."  He  finds  that  the  relation  of  the 
velocities  /,  q,  r,  in  the  above  example,  is  '^pxx  + 
pzz  —pyy  —  2aaq  —  2yxq  +  2zyq  +  2zxr  -\-yyr  •—  ^zzr  =  o. 
In  proving  his  rules  for  differentiation,  Newton 
divides  by  o^  and  in  the  resulting  expression  observes 
that  ''those  terms  in  which  o  is,  are  infinitely  less 
than  those  in  which  it  is  not.  Therefore,  blotting 
them  out,  there  rests  "  the  relation  sought.  The 
notation  by  dots,  "  pricked  letters,"  occurs  on  a  leaf, 
dated  May  20,  1665,  which  has  never  been  printed.  ^ 

49.  It  is  evident  that  Newton  permitted  twenty- 
eight  years  to  pass  between  the  time  of  his  first 
researches  on  fluxions  and  1693,  the  date  when  the 
earliest  printed  account  of  his  notation  of  fluxions 
appeared  from  his  pen  in  the  Latin  edition  of  Wallis's 
Algebra.  Moments  and  fluxions  are  mentioned  in 
his  Principia^  as  has  been  shown  by  our  extracts. 

50.  Of  importance  in  the  interpretation  of  the 
meanings  of   "moment"  in   the   second   edition   of 

^  S.  P.  Kigaud,  op.  cif.f  Appendix,  p.  23.  Consult  also  the  remarks 
on  this  passage  macie  by  CJ,  Ene^tròm  in  Bibliotheca  maihe>natica, 
3.  F.,  Bd.  II,  Leipzig,  1910-191 1,  p.  276,  and  Bd.  12,  1911-1912, 
p.  268,  and  by  A.  Wiiiing  in  Bd,  12,  pp.  56-60.  See  also  A  catalogne 
of  the  l'ortsinouth  collection  of  books  aìid  papers^  written  by  or  belonging 
to  Isaac  Newton,  Cambridge,  1889. 


NEWTON  31 

the  Principia  (17 13)  is  a  letter  of  May  15,  17 14, 
from  Newton  to  KeilV  from  which  we  quote  the 
foUowing  : — 

51.  ".  .  .  altho  I  use  prickt  Letters  in  the  first 
Proposition  of  the  hook  of  Quadratures,  yet  I  do 
not  there  make  them  necessary  to  the  method. 
For  in  the  Introduction  to  that  hook  I  describe 
the  method  at  large  &  illustrate  it  w"'  various 
examples  without  making  any  use  of  such  letters. 
And  it  cannot  be  said  that  when  I  wrote  that 
Preface  I  did  not  understand  the  method  of  fluxions 
because  I  did  not  there  make  use  of  prickt  letters 
in  solving  of  Problems.^  The  book  of  Quadratures 
is  ancient,  many  things  being  cited  out  of  it  by  me 
in  my  Letter  of  24  Octob.   1676.    .   .   . 

52.  ''ffluxions  &  moments  are  quantities  of 
a  different  kind.  ffluxions  are  finite  motions, 
moments  are  infinitely  little  parts.  1  put  letters 
with  pricks  for  fìiuxions,  &  multiply  fluxions  by  the 
letter  0  to  make  them  become  infinitely  little  and 
the  rectangles  I  put  for  moments.  And  wherever 
prickt  letters  represent  moments  &  are  without  the 
letter  o  this  letter  is  always  understood.  Wherever 
•^>  j)  y^  7)  ^tc,  are  put  for  moments  they  are  put  for 
xo^  yo^  j/00,  yo^.  In  demonstrating  Propositions  I 
always  write  down  the  letter  0  &  proceed  by  the 
Geometry  of  Euclide  and  ApoUonius  without  any 

^  J.    Edleston,    Correspondence  of  Sir  Isaac  A/ewlon    and  Professor 
CoUs,  London,  1850,  pp.  176,  177. 

yji^  John  Bernoulii,  in  tlie  Ada  Ertiditorum  for  February  and  March, 
17 13,  had  critici-sed  a  passage  in  the  Principia^  and  claimed  ihal  Newton 
did  not  understand  the  second  fluxions  when  writinij  that  passagc. 


32  LIMITS    AND   FLUXIONS 

approximation.  In  resolving  Ouestions  or  investi- 
gating  truths  I  use  ali  sorts  of  approximations 
w^''  1  think  will  create  no  error  in  the  conclusion 
and  neglect  to  write  down  the  letter  o,  and  this  do 
for  making  dispatch.  But  vvhere  x,  j>,  j/,  y  are  put 
for  fluxions  without  the  letter  o  understood  to  make 
them  infinitely  little  quantities  they  never  signify 
differences.  The  great  Mathematician  ^  therefore 
acts  unskilfuUy  in  comparing  prickt  letters  with 
the  marks  dx  and  dy^  those  being  quantities  of  a 
different  kind.  " 

Remarks 

53.  The  extracts  from  Newton's  writings  demon- 
strate  the  following  : — 

(i)  At  first  Newton  used  infinitesimals  (infinitely 
small  quantities),  as  did  Leibniz  and  other  mathe- 
maticians  of  that  age.  As  early  as  1665,  when 
Newton  was  a  young  man  of  twenty-three,  he  used 
them  and  speaks  of  "  blotting  them  out."^  He 
uses  infinitesimals  in  the  Principia  of  1687^  and  in 
his  account  of  the  quadrature  of  curves  in  Wallis's 
Algebra  of  1693,  where  Newton  speaks  of  himself 
in  the  third  person.^  It  is  worthy  of  emphasis,  in 
contrast  to  Leibniz,  that  Newton  uses  only  infinitesi- 
mals of  the  first  order.  Moreover,  as  De  Morgan 
remarked  long  ago,^  ''the  early  distinction  between 
the  systems  of  the  two  is  this,  that  Newton, 
holding  to  the  conception  of  the  velocity  or  fluxion, 

'  John  Bernoulli.     See  Edleston,  op.  cit.,  p.  171.  ~  See  our  §  48. 

3  See  our  §§  io,  13,  16,  18.  ^  See  our  §§  21,  26. 

^  De  Morgan,  Philosophtcal  Alagaztne,  4  S.,  voi.  iv,  1852,  p.  324. 


NEWTON  33 

used  the  infinitely  small  increment  as  a  means  of 
determining  it  ;  while,  with  Leibnitz,  the  relation 
of  the  infinitely  small  increments  is  itself  the  object 
of  determination.  " 

(2)  As  early  as  1665,  Newton  speaks  of  describing 
an  ''infinitely  little  line"  in  "  one  moment,"  and 
then  uses  the  expression  "in  the  next  "  moment.^ 
Here  **  moment"  cannot  mean  a  point  of  time, 
destitute  of  duration  ;  it  means  an  infinitely  small 
duration,  an  infinitesimal  of  time.  Doubtless  this 
use  of  "moment"  with  reference  to  time  suggested 
the  more  extended  and  general  use  of  the  term 
"  momentum  "  or  "  momenta  "  as  found  in  the 
Principia  ^  and  later  publications. 

(3)  The  use  of  dots,  "  prickt  letters,"  to  indicate 
velocities  or  fluxions  goes  back  to  1665,^  but  they 
are  not  used  by  Newton  in  print  until  1693  ^" 
Wallis's  Algebra  ;  they  are  used  extensively  in 
Newton's   Quadrature  of  Curves  of  1704.* 

(4)  Newton  first  used  the  word  "  fluxion  "  in 
print  in    1687   in  the  Principiai 

(5)  The  first  refinement  of  the  doctrine  of  fluxions 
is  found  in  Newton's  Principia,  where  he  speaks  of 
"prime  and  ultimate  ratios"^  and  of  "limits. "^ 

(6)  The  high-water  mark  of  Newton's  efforts  to 
place  the  doctrine  of  fluxions  upon  a  thoroughly 
logicai  basis  is  found  in  his  Quadrature  of  Curves, 
1704.      It  indicates  the  almost  complete  exclusion 

^  See  our  §  48.  -  See  our  §§  16,  18,  21,  24. 

^  See  our  §  48.  *  See  our  §§  22,  25,  44,  45. 

•"'  See  our  §§  16,  18.  «  See  our  §§  io,  13. 

'  See  our  §§  4,  6,  8,  io,  13. 


34  LTMITS    AND   FLUXIONS 

of  quantities  infinitely  little.  "  1  consider  mathe- 
matical  quantities  in  this  place  not  as  consistiiig  of 
very  small  parts, "  says  Newton.^  Also  "  the  very 
smallest  errors  in  mathematica!  matters  are  not  to 
be  neglected, "-  and  "in  the  method  of  fluxions 
there  is  no  necessity  of  introducing  figures  infinitely 
small  into  geometry.  "^  In  view  of  these  statements 
the  symbol  o  used  in  the  Quadrature  of  Curves,  a 
''quantitas  ad  modum  parva,"*  must  be  interpreted 
as  a  small  finite  quantity.  In  this  connection  De 
Morgan's  remarks  are  of  interest  :  ^  "In  1704, 
Newton  in  the  Quadratura  Cuì'vai'um  renounced 
and  abjured  the  infinitely  small  quantity  ;  but  he 
did  it  in  a  manner  which  would  lead  any  one  to 
suppose  that  he  had  never  held  it.  .  .  .  And  yet, 
there  is  something  like  a  recognition  of  some  one 
having  used  infinitely  small  quantities  in  Fluxions, 
contained  in  the  following  words  :  volui  ostendere 
quod  in  Methodo  Fluxionum  non  opus  sit  figuras 
infinite  parvas  in  Geometriam  introducere  :  nothing 
is  wanted  except  an  avowal  that  the  some  one  vvas 
Newton  himself  The  want  of  this  avowal  was 
afterwards  a  rock  of  offence.  Berkeley,  in  the 
Analyst,  could  not  or  would  not  see  that  Newton 
of  1687  ^^""d  Newton  of  1704  were  of  two  different 
modes  of  thought.  " 

We    do    not    interpret    Newton's    expressions   of 
1704   as  declarations   that   a   logicai    exposition    of 

1  See  our  §§  27,  34.  2  See  our  §§  30,  39. 

»  See  our  §§  33,  41.  "  See  our  §§  44,  45. 

•■*  De     Morgan,     Philosophical  Magazine,     4    S.,     voi.     iv.     1852, 
p.   328. 


NEWTON  35 

fluxions  cannot  be  given  on  the  basis  of  infini- 
tesimals  or  that  infinitely  small  quantities  are 
impossible  ;  for  he  says/  ''the  analysis  may  be 
performed  in  any  kind  of  figures  whether  finite  or 
infinitely  small,  which  are  imagined  similar  to  the 
evanescent  figures." 

In  fact,  not  even  in  1704  did  Newton  succeed  in 
completely  banishing  from  his  doctrine  of  fluxions 
the  infinitely  little.  If  what  he  used  in  1704  is 
not  the  infinitely  little,  it  is  so  closely  related  thereto, 
that  it  cannot  be  called  either  a  finite  magnitude  or 
an  absolute  zero. 

In  1704,  fluxions  are  "in  the  fif-st  ratio  of  the 
nascent  augments,  "  or  '  '  in  the  ultimate  ratio  of  the 
evanescent  parts.  "  ^  Unless  the  fully  developed 
theory  of  limits  is  read  into  these  phrases,  they 
will  involve  either  infinitely  little  parts  or  other 
quantities  no  less  mysterious.  At  any  rate,  the 
history  of  fluxions  shows  that  these  expressions 
did  not  meet  the  demands  for  clearness  and  freedom 
from  mysticism.  Newton  himself  knew  full  well 
the  logicai  difiiculty  involved  in  the  words  "prime 
and  ultimate  ratios  "  ;  for  in  1687  he  said,^  "it  is 
objected,  that  there  is  no  ultimate  proportion  of 
evanescent  quantities  ;  because  the  proportion, 
before  the  quantities  have  vanished,  is  not  ultimate  ; 
and,  when  they  have  vanished,  is  none."  How  does 
Newton  meet  this,  his  own  unanswerable  argument  ? 
He  does  so  simply  by  stating  the  difificulty  in  another 

1  See  our  §§  33,  42.    '^  See  our  §§  29,  30,  33,  36,  i%,  39,  42. 
'  See  our  §§  u,  14. 


36  LIMTTS   AND   FLUXIONS 

form  :  "  But,  by  the  same  argument,  it  might  as 
well  be  maintained,  that  there  is  no  ultimate  velocity 
of  a  body  arriving  at  a  certain  place,  when  its  motion 
is  ended  :  because  the  velocity,  before  the  body 
arrives  at  the  place,  is  not  its  ultimate  velocity  ; 
when  it  has  arrived,  is  none.  But  the  answer  is 
easy  :  for  by  the  ultimate  velocity  is  meant  that 
.  .  .  at  the  very  instant  when  it  arrives."  If 
''instant,"  as  used  here,  is  not  an  infinitesimal,  the 
passage  would  seem  to  be  difficult  or  impossible  of 
interpretation. 

(7)  A  return  to  the  open  use  of  the  infìnitely 
small  quantities  is  seen  in  writings  of  Newton  after 
the  year  1704.  It  might  be  argued  that  such  a 
return  was  necessary  in  the  second  edition  of  the 
Principia,  17 13,  unless  the  work  were  largely  re- 
written.  Newton's  Anafysis  per  cequationes  numero 
terniinorum  infinitas  was  first  printed  in  171 1,  and 
might  bave  been  rewritten  so  as  to  exclude  infini- 
tesimals  as  fully  as  was  done  in  the  Quadrature  of 
Curves  of  1704.  But  the  infìnitely  little  is  per- 
mitted  to  remain.^  There  is  no  disavowal  of  such 
quantities  either  in  the  Commercium  Epistolicum, 
with  the  editors  of  which  Newton  was  in  touch, 
or  in  Newton's  own  account  of  this  publication, 
contributed  to  the  Pìiilosophical  Transactions.^ 

(8)  The  theory  of  limits  is  involved  in  the  first 
lemma  of  the  Principia,^  and  in  the  explanation  of 
prime  and  ultimate  ratios  as  given  in  that  work. 

1  See  our  §  66.  -  See  our  §  47. 

3  See  our  §§  4,  6,  8,  9,  io,  12,  13,  15. 


CHAPTER  li 

PRINTED  BOOKS  AND  ARTICLES  ON  FLUXIONS 
BEFORE  1734 

54.  The  earliest  printed  publication  in  Great 
Britain  on  the  new  calculus  was  from  the  pen  of 
John  Craig,  a  Scotsman  by  birth,  who  settled  in 
Cambridge  and  became  a  friend  of  Newton.  Later 
he  was  rector  of  Gillingham  in  Dorsetshire.  He 
was  "  an  inoffensive,  virtuous  man,"  fond  of  mathe- 
matics.  In  1685  he  published  at  London  a  book 
entitled,  Methodus  figuraruin  .  .  .  quadraiuras 
determinandi.  At  that  time  nothing  could  be 
known  about  fluxions  except  through  private  com- 
munication.  In  1684  Leibniz  published  his  first 
ideas  of  Differential  Calculus  in  the  Leipzig  Acts. 
Craig  used  in  1685  the  calculus  of  Leibniz  and  also 
the  notation  of  Leibniz.  Continental  writers  cali 
Craig  the  introducer  of  the  theory  of  Leibniz  into 
England.  On  p.  28  of  his  book,  Craig  derives 
dp  =  4Snr*jy^d_y  from  p=  lónr^jy^,  and  arrives  at  a 
differential  equation  (aiquationem  differentialem). 
The  meanings  of  <^,  dy,  dx,  etc. ,  are  not  explained 
but  taken  for  granted,  reference  being  made  to 
Leibniz.  In  1693  Craig  published  another  book  in 
which   the    notation  of  Leibniz   is   used.      He  con- 

37 


38  LIMITS    AND   FLUXIONS 

tributed  also  several  papers  to  the  Philosophical 
Trans act io ns  {l^ondon),  but  never,  before  17 18,  did 
he  use  fluxional  symbols.  In  preparing  the  book 
of  1685  he  had  received  from  Newton  the  binomial 
theorem  which  he  used  before  it  had  appeared  in 
print,  but  he  had  no  communication  about  fluxions. 
''We  have  here  the  singular  indifference,  "  says  De 
Morgan,  "which  Newton  at  that  time,  and  long 
afterwards,  showed  toward  his  own  calculus."^ 
Craig  wrote  a  tract  in  1693,  and  articles  for  the 
Philosophical  Transactzons  \niy  01,  1703,  1704,  1708, 
using  the  differential  calculus  ali  this  time.  In  the 
issue  No.  284,  1703,  he  employs  the  Leibnizian 
sign  of  integration  /.  Craig  submitted  to  Newton 
one  of  his  early  manuscripts  (probably  the  one 
printed  in  1693).  With  regard  to  this  event  De 
Morgan  wrote  to  Hamilton,  the  inventor  of  quater- 
nions  :  *'  Few  of  us  know  that  Leibniz  was  perfectly 
well  known  in  England  before  the  dispute,  and  that 
Newton's  first  provocative  to  an  imperfect  publica- 
tion  was  ds  and  infinitely  small  quantities  paraded 
under  his  own  eyes  by  an  English  writer  (Craig), 
who  lent  him  his  MSS.  to  read."^  Craig's  publica- 
tion  of  17 18  foUowed  the  great  controversy  on  the 
invention  of  the  calculus  ;  now  he  uses  fluxions 
exclusively  and  says  not  a  word  on  the  differential 
calculus.  The  book  does  not  discuss  fundamentals, 
and  no  explanation  of  x  is  given.      As  conjectured 

^  De  Morgan,  Philosophùaì  Magazine,  4  S.,  voi.  iv,  1S52,  p. 
326. 

'^  Life  of  Str  IVilliam  Rowan  Hamilton,  by  Robert  F.  Graves, 
voi.  iii,  1889,  p.  415. 


PRINTED  BOOKS,   ETC,  BEFORE   1734     39 

by  De  Morgan,  it  may  have  been  Craig's  manu- 
script  that  suggested  to  Newton  the  need  of  making 
bis  own  fluxions  accessible  to  the  public.  At  any 
rate,  in  1693  there  appeared  the  account  of  fluxions 
in  Wallis's  Algebra.      [See  Addenda,  p.  289.] 

55.  Abraham  De  Moivre,  a  French  mathemati- 
cian  who  in  1688,  after  the  revocation  of  the  Edict 
of  Nantes,  carne  to  London,  contributed  in  1695  to 
No.  216  of  the  Philosophical  Transactions  (London) 
an  article  in  which  he  uses  x,  j>,  x,  y,  and  lets  both 
''fluxion"  and  "moment"  stand  for  things  infin- 
itely  small.  In  the  same  number  of  the  Transac- 
tions, the  astronomer  Edmund  Halley  has  an  article 
on  logarithms  in  which  he  uses  infinitely  small 
ratiunculce  and  differentiolce,  but  neither  the  nota- 
tion  of  Leibniz  nor  that  of  Newton.  In  1697,  David 
Gregory  used  in  No.  231  of  the  Transactions  x  ^nd 
speaks  of  "  fluxio  fluxionis  "  without,  however,  ex- 
plaining  bis  terms. 

56.  Patio  de  Duillier,  a  Swiss  by  birth,  who  had 
settled  in  London  and  become  member  of  the 
Royal  Society,  wrote  in  1699  a  treatise,  Lincee 
brevissimi  dcscensus  investigatio  geometrica,  uses 
fluxions  as  infinitely  small  quantities.  This  publi- 
cation  is  noted  as  containing  a  statement  which 
started  the  Newton-Leibniz  controversy  on  the 
invention  of  the  Calculus. 

57.  It  is  remarkable  that  Roger  Cotes,  in  1701, 
when  an  undergraduate  at  Trinity  College,  Cam- 
bridge [Newton's  own  College],  wrote  a  letter  on 
mathematica!    subjccts,    in    which   x    is    used    as 


40  LIMITS    AND   FLUXIONS 

''infinitely  little."'  In  1702-3  Humphry  Ditton, 
in  voi.  xxiii  of  the  Transactions ^  used  the  fluxional 
notation,  without  explanation. 

58.  Other  vvritings  that  do  not  define  their  terms 
are  the  Fluxionum  rnethodus  inversa,  1704,  by  the 
London  physician,  George  Cheyne,  and  De  Moivre's 
Animadvei'siones  in  D.  Georgii  Cheynai  Tractaium, 
London,  1704.  However,  Cheyne  lets  x—  i,  from 
which  we  infer  that,  with  him,  x  was  finite.  [See 
Addenda,  p.  289.] 

59.  The  next  writer  on  fluxions  was  John  Harris, 
a  voluminous  author  of  books  on  various  subjects. 
He  was  at  one  time  Secretary  of  the  Royal  Society. 
In  1702  he  published  at  London  A  New  Short 
Treatise  of  Algebra,  which  devotes  the  last  22 
pages,  out  of  a  total  of  136  pages,  to  fluxions.  It 
is  the  first  hook  in  the  English  language  in  which 
this  subject  is  treated.  The  doctrine  of  fluxions  is 
the  "  Arithmetick  of  the  I n finii  e  ly  s  mail  Increments 
or  Decrements  of  Indeterminate  or  Variable  Qua?i- 
tities,  or  as  some  cali  them  the  Moments  or  hifin- 
itely  small  Differences  of  such  Variable  Ouantities. 
These  Infinitely  small  Increments  or  Decrements, 
our  incomparable  Mr.  Isaac  Newton  calls  very  pro- 
perly  by  this  name  of  Fluxions  "  (p.  115).  A  few 
lines  further  on  it  says  that  Newton  ''calls  the 
celerity  or  Velocity  of  the  Augmentation  of  Diminu- 
tion  of  these  Flowing  Quantities,  by  the  name  of 
Fluxions.''  A  second  edition  of  this  hook  appeared 
in    1705.      As  authors  on  fluxions,  Harris  in    1705 

^  J.  Edleston,    Correspondence  of  Sir  Isaac   Newton   and  Professor 
Cotes,  London,  1850,  p.  196. 


PRTNTED  BOOKS,   ETC,   BEFORE   1/34     4i 

mentions  Newton,  Wallis,  Nieuwentiit,  Carré,  Leib- 
niz, l'Hospital,  de  Moivre,  and  Hayes. 

60.  John  Harris  also  published  a  Lexicon  Tech- 
nìcuin,  of  which  the  second  volume,  London,  17 io, 
contains  an  article,   "Fluxions." 

"  This  general  Method  of  finding  the  Fluxions  of 
ali  Powers  and  Roots,  I  had  from  the  Hon.  Fr. 
Robartes,  Esq.  If  a  Ouantity  gradually  increases 
or  decreases,  its  immediate  Increment  or  Decre- 
ment  is  called  its  Fliixion.  Or  the  Fluxion  of 
a  Quantity  is  its  Increase  or  Decrease  indefinitely 
small.  .  .  .  Since  xx  .  .  .  is  infinitely  smaller  than 
2xx^  whereby  it  can  make  no  sensible  Change  in 
that  Quantity,  it  may  be  laid  aside  as  of  no  Value. 
.  .  .  Authors' Nameswho  have  written  of  Fluxions: 
D.  Bernoulli  Ti-actatus  de  Principiis  Calculi  Exponen- 
tialis\  Nieuwentiifs  Analysis  Infinitoi-um^  Amster. , 
1695  ;  Dr.  Cheyne's  Fluxions^  with  Moivre' s  Anim- 
adversiotis  on  them,  and  the  Doctor's  reply  ;  Hays's 
Fluxions,  Lond.,  1704;  Analyse  des  Infiniinent 
Petits.  Par  l'Hospital,  Fr.,  Paris,  1696;  Le  Calcule 
Integrale,  par  M.  Carré,  Paris,  1700;  Mr.  Abraham 
de  Moivre's  Use  of  Fluxions,  in  the  Solution  of 
Geometrick  Problems.  See  Philos.  Trans.,  N.  216; 
Mr.  Humphry  Ditton's  Institution  of  Fluxions.  " 

61.  In  the  above  list  of  writers  are  Charles 
Hayes  and  Humphry  Ditton,  authors  of  English 
texts  now  demanding  our  attention.  Hayes  starts 
his  elucidation  of  fundamentals  (p.   i)  as  follows  :^ 

^  A    Treafisc  of  F/uxions  :   or,   An   Introduction    to  Afathematical 
Philosophy,  Charles  Hayes,  London,  1704. 


42  LIMITS    AND   FLUXIONS 

"  Magnitude  is  divisible  in  infinitum,  and  the 
Parts  after  this  infinite  Division,  being  infinitely 
little,  are  what  Analysts  cali  Moments  or  Differ- 
ences  ;  And  if  we  consider  Magnitude  as  Indeter- 
minate and  perpetually  Increasing  or  Decreasing, 
then  the  infinitely  little  Increment  or  Decrement  is 
call'd  the  Fluxion  of  that  Magnitude  or  Quantity  : 
And  whether  they  be  called  Moments,  Differences 
or  Fluxions,  they  are  stili  suppos'd  to  have  the 
same  Proportion  to  their  Whole's,  as  a  Finite 
Number  has  to  an  Infinite  ; 
or  as  a  finite  Space  has  to 
an  infinite  Space.  Now  those 
infinitely  little  Parts  being 
extended,  are  again  infinitely 
Divisible  ;  and  these  infinitely 
little  Parts  of  an  infinitelylittle 
Part  of  a  given  Ouantity,  are 

tlG.   3.  O  ^  J  1 

by  Geometers  call'd  Injìnite- 
siince  Infinitesiììiaruiìi  or  Fluxions  of  Fluxìons. 
Again,  one  of  those  infinitely  little  Parts  may  be 
conceiv'd  to  be  Divided  into  an  infinite  Number  of 
Parts  which  are  call'd  Third  Fluxions,  etc.  " 

He  endeavours  to  justify  this  doctrine  by  illus- 
trations.  The  angle  of  contact  FAG  formed  by  the 
line  AE  and  the  ordinary  parabola  AG,  is  less  than 
any  rectilineal  angle  ;  the  angle  P'AD,  formed  by 
AE  with  the  cubical  parabola  AD,  is  infinitely 
less  than  the  angle  FAG,  and  so  on.  Hayes 
defines  the  doctrine  of  Fluxions  as  the  "  Arith- 
metick  of  infinitely  small  Increments  or  Decrements 


PRINTED  BOOKS,  ETC,  BEFORE  1734     43 

of  Indeterminate  or  variable  Quantities."  He 
cautions  the  reader  :  "  But  we  must  take  great 
heed,  not  to  consider  the  Fluxions,  or  Increments, 
or  Decrements  as  finite  Quantities  "  (p.  4).  He 
rejects  xèy  and  xèy  ''as  being  incomparably  less  " 
than  xzy. 

The  same  year  in  which  Hayes  wrote  this  first 
English  hook  on  fluxions  which  could  make  any 
claim  to  attention,  saw  the  appearance  of  Newton's 
Quadratuì-a  Curvarum.  The  contrast  in  the  defini- 
tion  of  "  fluxion  "  was  sharp.  Hayes  called  it  "  an 
infinitely  small  increment  "  ;  Newton  called  it  a 
"velocity,  "  a  finite  quantity. 

62.  William  Jones,  in  his  Synopsis  P almariot-uni 
Matheseos,  London,  1706,  devotes  a  few  pages  to 
fluxions  and  fluents,  using  the  Newtonian  notation. 
On  p.  225  he  gives,  in  substance,  Newton's  lemma, 
in  these  words  :  "Quantities,  as  also  their  Ratio's, 
that  continually  tend  to  an  Equality,  and  therefore 
that  approach  nearer  the  one  to  the  other,  than 
any  Difìference  that  can  possibly  be  assign'd,  do 
at  last  become  equal."  Then  he  says  :  "Hence  ali 
Curved  Lines  may  be  considered  as  composed  of 
an  Infinite  Number  of  Infinitely  little  right  Lines." 
He  uses  "infinitely  small"  quantities,  but  defines 
a  fluxion  as  "the  Celerity  of  the  Motion,"  fluxions 
being  "in  the  first  Ratio  of  their  Nascent  Aug- 
ments."  Jones  represents  bere  the  Newton  of  the 
Principia,  and  of  the  Quadrature  of  Curves  as  given 
in  1793. 

63.  The  earliest   hook  exhibiting  a  careful  study 


44  LI  MIT  S    AND   FLUXIONS 

of  Newton's  tract  of  1704  was  Humphry  Ditton's 
Instiiution  of  Fluxions,  1706.^  Ditton  was  pro- 
minent  as  a  divine  as  well  as  a  mathematician. 
Like  so  many  other  Eiiglish  writers  on  fluxions 
during  the  eighteenth  century,  he  had  not  been  at 
either  of  the  great  universities.  He  states  in  his 
preface  that  he  has  also  consulted  and  drawn  from 
the  writings  of  John  Bernoulli  and  some  other 
Continental  writers. 

64.  The  reader  of  Ditton's  hook  is  impressed  by 
the  fact  that  he  labours  strenuously  to  make  every- 
thing  plain.  He  takes  the  reader  fully  into  his 
confìdence.  This  is  evident  in  the  extracts  which 
follow  (pp.  12-21)  : — 

"Suppose  any  flowing  Quantities,  .  .  .  as  also 
their  Increments  .  .  .  which  Increment  imagine  to 
be  generated  in  equal  very  small  Tarticles  of  Time. 
I  conceive  we  may  say  without  Scruple,  that  the 
Fluxions  are  ihe  velocities  of  those  Increments,  con- 
sideraci not  as  actually  genei'ated,  but  quatenus 
Nascentia,  as  arising  and  beginning  to  be  genei^ated. 
As  there  is  a  vast  difference  between  the  Increments 
consider'd  as  Finite,  or  really  and  actually  generated  ; 
and  the  same  considered  only  as  Nasce?ttia  or  in  the 
first  Moment  of  their  Generation  :  So  there  is  as 
great  a  difference  also  between  the  Velocities  of  the 
Increments,  consider'd  in  this  two  fold  respect.   .   .   . 

^  An  Institution  of  Fluxions  :  Containmg  the  First  Pnnciples,  The 
Operatiovs,  with  some  of  the  Uses  and  Applications  of  that  Admirable 
Method  ;  According  to  the  Schet/te  prefixd  to  his  Iract  of  Qiiadraturcs, 
by  {its  First  Inventor)  the  hicoviparable  Sir  Isaac  Newton.  By 
Humphry  Ditton,  London,  1706. 


PRINTED  BOOKS,  ETC,   BEFORE   1734     45 

The  Reason  of  that  [difference],  is  this.  Because 
there  is  (speaking  strictly  and  accurately)  an  Infinity 
of  Velocities  to  be  consider'd,  in  the  Generation  and 
Production  of  a  Real  hicrernent  ;  ...  So  that  if  we 
conceiv'd  the  Fluxion,  to  be  the  Velocity  of  the 
Increment,  as  actually  Generateci  ;  we  must  conceive 
it  to  be  an  Infinite  Variety  or  Series  of  Velocities. 
Whereas  the  Velocity,  with  which  any  sort  of  In- 
crement arises,  or  begins  to  be  generateli  ;  is  a  thing 
that  one  may  form  a  very  clear  and  distinct  Idea  of, 
and  leaves  the  Mind  in  no  Ambiguity  or  Confusion 
at  ali.  .  .  .  However,  if  we  take  those  Particles 
of  time  exceeding  small  indeed,  and  Neglect  the 
Acceleration  of  the  Velocity  as  inconsiderable,  we 
may  say  the  Fluxions  are  proportional  to  those  In- 
crements  ;  remembering  at  the  same  time,  that  they 
are  but  nearly,  and  not  accurately  so.  .  .  .  If  in  the 
Differential  Calculus,  some  Terms  are  reìected  and 
thrown  out  of  an  Equation,  because  they  are  nothing 
Comparatively,  or  with  respect  to  other  Terms  in  the 
same  Equation  ;  that  is,  because  they  are  infinitely 
small  in  proportion  to  those  other  Terms,  and  so  may 
be  neglected  upon  that  Score:  On  the  other  band,  in 
the  Method  of  Fluxions,  those  same  Terms  go  out 
of  the- Equation,  because  they  are  multiplied  into  a 
Ouantity,  which  .  .  .  does  at  last  really  vanish.  .  .  . 
N.B.  Speaking  bere  of  Infinitely  small  Quantities, 
or  Infinitesimals  as  some  Authors  (and  particularly 
Mr.  Neiwentiit)  chuse  to  term  them,  I  cannot  but 
take  notice  of  a  notion,  which  that  Excellent  and  In- 
genious  Ferson  advances  in  bis  Analysis  htjìnitorum. 


46  LIMITS    AND   FLUXIONS 

It  is  this  ;  That  a  Quantity  Infinitely  Great,  a  Finite  or 
any  given  Quantity,  an  Infìnitesimal,  a7id  Nihilum 
Geornetricum,  are  in  Geometrical  Troportion.  I 
confess  I  cannot  discover  the  truth  of  this.  .  .  . 
Let  in  denote  an  infinite  Quantity,  d  any  finite  one  ; 
then  is  d  /  ;;/  the  Infinitesimal  of  d,  according  to 
Mr.  Neiwentiit.  Now  his  Assertion  is,  that  m  :  d  :  : 
d  I  m  :  o  ;  therefore  since  from  the  nature  of  Geo- 
metrical Proportion,  'tis  aiso  m  :  d  :  :  d  /  m  :  dd  /  mm  ; 
it  foUows  that  dd  I  mm  is  =  o  .  .  .  then  d  /  m  =  o. 
Now  Mr.  Neiwentiit  will  hardly  allow  his  Infinitesimal 
to  be  nothing  ;  and  yet  ...  I  think  it  must  follow, 
that  d=o.''  Proceeding  geometrically,  Ditton  ex- 
plains  the  fluxions  of  lines,  areas,  solids,  and  surfaces. 
Next  he  takes  up  algebraical  expressions.  To  find 
the  fluxion  of  x'%  he  lets  x  flow  uniformly  and  re- 
presents  the  augment  of  ;r  in  a  given  particle  of  time 
by  the  symbol  o.  While  x  becomes  x-\-o,  x"  becomes 
(x-\-oy.  Expanding  the  binomial,  he  finds  that  the 
two  augments  are  as  i  to  nx''~'^-{-(n'^ —  n)ox'''^  /  2  + 
etc.  "  And  the  Ratio  of  them  (making  o  to  vanish) 
will  be  that  of  i  to  nx""'^."  According  to  his  nota- 
tion  i"  is  a  fluxion  of  x,  and  .ir  is  a  fluxion  of  a:. 
Taking  ^  as  a  very  small  quantity,  he  lets  the  ex- 
pressions oà,  oy  represent  the  moinents,  or  increments 
of  the  flowing  quantity  z,  y  generated  in  a  very  small 
part  of  time.  "  If  therefore  now,  at  the  present 
Moment,  the  flowing  Quantities  are  z,  y,  x  ;.  the  next 
Moment  (when  augmented  by  these  Increments) 
they  will  become  x+oò;  y-\-oy,  x-\-ox.''  He  ex- 
presses  the  general   mode  of  procedure  for  finding 


PRINTED  BOOKS,  ETC,  BEFORE   i734     47 

the  fluxion,  vvhich  coincides  with  the  modem  mode 
of  finding  a  derivative.  Ditton  considers  the  in- 
crements  as  finite  (p.  53).  "  These  Momenta  are  in 
proportion  to  one  another  as  the  Fluxions  of  the 
flowing  Quantities  respectively,  for  02,  oy,  ox,  are 
as  2,  j>,  i'  ;  and  Mr.  Newton  had  before  expresly 
told  US  ;  that  the  Increments  generated  in  a  very  small 
Farticle  of  time  were  very  nearly,  as  the  Fluxions." 
Evidently  Ditton  does  not  here  overlook  that 
02,  oy,  ox  represent  the  increments  only  *'very 
nearly."  He  observes  (p.  98)  that  \ve  may  "  go 
on  with  ease  to  the  second,  third,  and  any  other 
Fluxions  ;  neither  are  there  any  new  Difficulties  to 
be  met  with." 

A   second  edition   of  Ditton's  book  was  brought 
out  in  1726  by  John  Clarke. 

65.  Ditton's  first  edition  appeared  at  a  time  when 
the  Newton-Leibniz  controversy  was  under  way. 
Leibniz  had  appealed  to  the  Royal  Society  for 
justice.  That  Society  appointed  a  committee  which 
published  a  report  containing  letters  and  other 
material  hearing  on  the  case,  in  a  book  called  the 
Commercium  Epistolicum,'^  which  figures  prominently 
in  the  lamentable  controversy.  From  this  book  the 
early  use  of  infinitely  small  quantities  on  the  part 
of  Newton  is  conspicuously  evident.  The  book 
makes  it  clear  also  that  some  of  Newton's  warmest 
supporters  were  guilty  of  gross  inaccuracy  in  the 
use  of  the  word  "fluxion." 

'   Commercium  Epistolicuìn  D.  /ohannis  Collins,  et  aliortttn  de  analysi 
promota  :  jussu  societatis  regia  in  lucevi  editum.     Lendini,  MDCCXII. 


48  LIMITS    AND   FLUXIONS 

66.  Newton's  Analysis  per  cequationes  numero 
terminonim  infinitas,  which  was  sent  on  July  31, 
1669,  through  Barrow  to  Collins,  and  which  was 
first  published  at  London  in  171 1,  was  reprinted  in 
the  Commercium  Epistolicuni.  In  this  Analysis  in- 
finitely  small  quantities  are  used  repeatedly,  but  the 
word  '  '  fluxion  "  and  the  fluxional  notation  do  not 
occur.  In  a  letter  to  H.  Sloane,  who  was  then 
Secretary  of  the  Royal  Society  of  London,  written 
in  answer  to  a  letter  of  Leibniz  dated  March  4,  17 1 1, 
John  Keill,  professor  of  astronomy  at  Oxford,  re- 
counts  the  achievements  of  Isaac  Barrow  and  James 
Gregory,  and  says  :  "  If  in  place  of  the  letter  0, 
which  represents  an  infinitely  small  quantity  in 
James  Gregory's  Geometrice  pars  uìiiversaìis  (1667), 
or  in  place  of  the  letters  a  ox  e  which  Barrow  em- 
ploys  for  the  same  thing,  we  take  the  x  or  y  of 
Newton  or  the  dx  or  dy  of  Leibniz,  we  arrive  at  the 
formulas  of  fluxions  or  of  the  differential  calculus."^ 
Thus  Keill,  the  would-be  great  champion  of  Newton, 
instead  of  warning  the  reader  against  confusing 
differentials  and  fluxions,  himself  comes  dangerously 
dose  to  conveying  the  erroneous  idea  that  x  and  y 
are  infinitely  small,  the  same  as  dx  and  dy.  He 
Comes  so  near  to  this  as  to  be  guilty  of  lack  of 
caution,  if  not  of  inaccuracy. 

More  serious  is  a  statement  further  on.      The  en- 


^  "  Nam  si  prò  Litera  0,  quae  in  Jacobi  Gregorii  Parte  Matheseos  Uni- 
versali quantitaleni  infinite  parvam  repraesentat  ;  aut  prò  Literis  a  vel  e 
quas  ad  eandem  designandam  adhibet  Barrovius  ;  ponamus  x  vel  y 
Newtoni,  vel  dx  seu  dy  Leibnitii,  in  P'ormulas  Fliixionum  vel  Calculi 
Differentialis  incidemus  "  (p.  112). 


PRINTED  BOOKS,  ETC,  BEFORE  1734     49 

listment  of  the  services  of  a  clever  lawyer  would  be 
needed  to  acquit  the  editors  of  the  Conimercium 
Epistolicum  of  gross  error  when,  in  the  final  summary 
of  their  case  against  Leibniz,  they  declare  (p.  121), 
"that  the  Differential  Method  is  one  and  the  same 
with  the  Method  of  Fluxions,  excepting  the  name  and 
the  notation  ;  Mr.  Leibniz  calling  those  Quantities 
Uifferences,  which  Mr.  Newton  calls  Moments  or 
Fluxions  ;  and  marking  them  with  the  letter  d^  a 
mark  not  used  by  Newton." 

òj .  Joseph  Raphson,  in  bis  History  of  Fluxions 
(which  appeared  as  a  posthumous  work  at  London, 
in  171 5,  printed  in  English,  and  in  the  same  year 
also  in  Latin,  the  Latin  edition  containing  new  corre- 
spondence  hearing  on  the  Newton-Leibniz  contro- 
versy),  says  on  p.  5  that  Newton  '  '  makes  use  of 
Points,  and  denotes  those  first  Differences  (which  by 
a  Name  congruous  to  their  Generation,  being  con- 
sider'd  as  the  first  Increments  or  Decrements  of  a 
continued  Motion,  he  calls  Fluxions)  thus,  viz.  x,  j>, 
-c.  "  This  misrepresentation  of  Newton  is  the  more 
astonishing  when  we  recollect  that  Raphson  was 
very  partial  to  Newton,  and  also  meant  his  History 
**to  open  a  plain  and  easy  way  for  Beginners  to 
understand  these  Matters.  "  Newton  never  looked 
upon  a  fluxion  as  anything  different  from  velocity  ; 
with  him  it  was  always  a  finite  quantity.  To  make 
matters  worse,  Raphson  continues  :  "To  these 
Quantities  he  adds  others  of  another  Gender,  and 
which  in  relation  to  Finite  ones  may  be  conceiv'd  as 
infinitely  great,  and   denotes   them  thus  'x,    'y,    'z, 

4 


50  LIMITS    AND   FLUXIONS 

whereof  the  first  or  finite  Ouantities  themselves,  viz. 
X,  V,  z,  may  be  conceiv'd  as  Fluxions.''  And  again, 
"a  Point  .  .  .  may  be  consider'd  as  the  Fluxion  of 
a  Line,  a  Line  as  the  Fluxion  of  a  Piane,  and  a 
Piane  as  the  Fluxion  of  a  Solid,  and  a  finite  Solid 
as  the  Fluxion  of  a  (parfially)  infinite  one,  and  that 
again  as  the  Fluxion  of  one  of  an  higher  Gender  of 
Infinity,  and  so  on  ad  inf.  which  we  shall  further 
illustrate  in  some  Dissertations  at  the  end  of  this 
Treatise." 

68.  Brook  Taylor  brought  out  at  London  in  17 15 
his  Methodus  incrementonim  dire  età  et  inversa,  in 
which  he  looks  upon  fluxions  strictly  from  the  stand- 
point  of  the  Newtonian  exposition  in  the  Quadrature 
of  Curves,   1704. 

69.  James  Stirling  uses  x  and  y  as  infinitesimals 
in  his  LinecE  tertii  ordinis,  Oxford,  17 17.  He  draws 
the  inftnitely  small  right  triangle  at  the  contact 
of  a  curve  with  its  asymptote,  the  horizontal  side 
being  "  quam  minima"  and  equal  to  i-,  the  vertical 
side  being  y.  In  the  appendix  to  this  booklet  of 
17 17,  X  and  y  are  again  infinitely  small.  In  his 
Methodus  differentialis,  London,  1730,  there  is  no 
direct  attempt  to  explain  fundamentals,  any  more 
than  there  was  in  17 17,  but  on  p.  80  he  puts  the 
fluxion  of  an  independent  variable  equal  to  unity, 
from  which  we  infer  that  a  fluxion  is  with  him  now 
a  finite  velocity. 

70.  For  twenty-four  years  after  Ditton  no  new 
text  appeared.  In  1730  Edmund  Stone,  a  self- 
taught  mathematician  who  had  studied  De  l'Hospital, 


PRINTED  BOOKS,   ETC.   BEFORE   1734     51 

sent  forth  a  new  book,  the  first  part  of  which  was 
a  translation.^ 

The  foUowing  extract  is  from  Stone's  translation 
of  De  l'Hospital's  Preface,  the  words  in  the  square 
brackets  [  ]  being  interpolateci  by  Stone  : — 

"  By  means  of  this  Analysis  we  compare  the 
infinitely  small  (DiiTerences  or)  Parts  of  finite 
Magnitudes,  and  find  their  Ratio's  to  each  other  ; 
and  hereby  Hkewise  learn  the  Ratio's  of  finite 
Magnitudes,  those  being  in  reality  so  many  infinitely 
great  Magnitudes,  in  respect  of  the  other  infinitely 
small  ones.  This  Analysis  may  ever  be  said  to  go 
beyond  the  Bounds  of  Infinity  itself  ;  as  not  being 
confined  to  infinitely  small  (Differences  or)  Parts, 
but  discovering  the  Ratio's  of  Differences  of  Differ- 
ences, or  of  infinitely  small  Parts  of  infinitely  small 
Parts,  and  even  the  Ratio's  of  infinitely  small  Parts 
of  these  again,  without  End.  So  that  it  not  only 
contains  the  Doctrine  of  Infinites,  but  that  of  an 
Infinity  of  Infinites.  It  is  an  Analysis  of  this  kind 
that  can  alone  lead  us  to  the  Knowledge  of  the  true 
Nature  and  Principles  of  Curves  :  For  Curves  being 
no  other  than  Polygons,  having  an  Infinite  Number 
of  Sides,  and  their  Differences  arising  altogether 
from  the  different  Angles  which  their  infinitely 
small  Sides  make  with  each  other,  it  is  the  Doctrine 
of  Infinites  alone  that  must  enable  us  to  determine 
the  Position   of  these    Sides,    in   order   to   get   the 

^  l^he  Method  of  Fluxions^  both  Direct  and  Inverse.  The  former  bein^ 
a  Translation  from  the  Celebrated  Marqiiis  De  PHospiiars  Analyse 
des  Injinenunts  Petits  :  And  the  Latter  Supply'd  by  the  Trans lator, 
E.  Stone,  F.R.S.  London,  MDCCXXX. 


52  LIMITS    AND   FLUXIONS 

Curvature  formed  by  them  ;  and  thence  the  Tangents, 
Perpendiculars,  Points  of  Inflexion  and  Retrogres- 
sion,  reflected  and  refracted  Rays,  etc. ,  of  Curves. 

'*Polygons  circumscribed  about  or  inscribed  in 
Curves,  whose  Number  of  Sides  infinitely  augmented 
till  at  last  they  coincide  with  the  Curves,  bave 
always  been  taken  for  Curves  themselves.  .  .  .  It 
was  the  Discovery  of  the  Ànalysis  of  Infinites  that 
first  pointed  out  the  vast  Extent  and  Fecundity  of 
this  Principle.  .  .  .  Yet  this  itself  is  not  so  simple 
as  Dr.  Barrow  afterwards  made  it,  from  a  dose 
Consideration  of  the  Nature  of  Pohgons,  which 
naturally  represent  to  the  Mind  a  Httle  Triangle 
consisting  of  a  Particle  of  a  Curve  (contained 
between  two  infinitely  near  Ordinates),  the  Differ- 
ence  of  the  correspondent  Absciss's  ;  and  this 
Triangle  is  similar  to  that  formed  by  the  Ordinate, 
Tangent,  and  Subtangent.  .  .  .  Dr.  Barrow  .  .  .  also 
invented  a  kind  of  Calculus  suitable  to  the  Method 
{Lect.  Geoin.,  p.  80),  tho'  deficient.  .  .  .  The 
Defect  of  this  Method  was  supplied  by  that  of 
Mr.  Leibnitz'z,^  [or  rather  the  great  Sir  Isaac 
Newton].'^  He  began  where  Dr.  Barrow  and  others 
left  off:  His  Calculus  has  carried  him  into  Countries 
hitherto  unknown.  ...  I  must  bere  in  justice  own 
(as  Mr  Leibnitz  himself  has  done,  in  Journal  dcs 
S^avans  for  August  1694)  that  the  learned  Sir  Isaac 
Newton  likewise  discover'd  something  like  the 
Calculus  Differentialis,  as  appears  by  his  excellent 

A  Ada  Eruàit.  Lips.,  ann.  1684,  p.  467. 
2  See  Commcrcùim  Epistoluuni. 


PRINTED  BOOKS,   ETC,  BEFORE  1734     53 

Principia^  published  first  in  the  Year  1687,  which 
almost  vvhoUy  depends  upon  the  Use  of  the  said 
Calculus.  But  the  Method  of  Mr.  Leibnitz'z  is 
much  more  easy  and  expeditious,  on  account  of 
the   Notation  he  uses.    .   .   ." 

In  the  preface  of  "  The  Translator  to  the  Reader  " 
Stone  points  out  that  the  work  he  is  bringing  out 
'*becomes  the  more  necessary,  because  there  are 
but  two  EngHsh  Treatises  on  the  Subject  .  .  .  the 
one  being  Hay's  Introduction  to  Mathematical 
Philosophy,  and  the  other,  Ditton's  Institutio7i  of 
Fluxions''  \  the  former  "  too  prolix,"  the  latter 
"much  too  sparing  in  Examples  "  and  "  too  re- 
dundant  "  in  the  explanation  of  fluxions,  so  that  "  it 
is  next  to  impossible  for  one  who  has  not  been 
conversant  about  Infinites  to  apprehend  it.  That 
of  our  Author  is  much  easier,  tho  less  Geo- 
metrica!, who  calls  a  Differentia]  (or  Fluxion)  the 
infinitely  small  Part  of  a  Magnitude.  "  "  But,"  con- 
tinues  Stone,  '  '  I  would  not  bere  be  thought  in  any 
wise  to  lessen  the  Value  of  Sir  Isaac  Newton's 
Definition  :  When  the  Learner  has  made  some 
Progress,  I  would  bave  him  then  make  himself 
Master  of  it."  Stone  then  proceeds  to  explain  the 
nature  of  fluxions,  foUowing  closely  Newton's 
language  in  his   Quadrature  of  Curves. 

71.  In  De  l'Hospital's  treatise,  as  translated  by 
Stone,  we  read  : 

"The  infinitely  small  Part  whereby  a  variable 
Quantity  is  continually  increased  or  decreas'd,  is 
called  the  Fluxion  of  that   Ouantity." 


54  LIMITS   AND  FLUXIONS 

Here  Stone  simply  writes  "  fluxìon  "  where 
De  l'Hospital  writes  "  difìerence,"  which  is  a 
mischievous  procedure,  seeing  that  the  two  words 
stand  for  things  totally  difìferent.  De  rHospital's 
wordiiig  is  *'La  portion  infiniment  petite  dont  une 
quantité  variable  augmente  ou  diminuè  continuelle- 
ment,  en  est  appellée  la  Difìférence.  "  Stone  also 
changes  from  the  Leibnizian  to  the  Newtonian 
notation,  by  writing  x  instead  of  dx.  Then  foUow 
two  postulates  : 

''Grant  that  two  Quantities,  whose  Difìférence 
is  an  infinitely  small  Ouantity,  may  be  taken  (or 
used)  indifferently  for  each  other  :  or  (which  is  the 
same  thing)  that  a  Ouantity,  which  is  increased  or 
decreas'd  only  by  an  infinitely  small  Quantity,  may 
be  consider'd  as  remaining  the  same. 

"  Grant  that  a  Curve  Line  may  be  consider'd 
as  the  Assemblale  of  an  infinite  Number  of  in- 
finitely small  right  Lines  :  or  (which  is  the  same 
thing)  as  a  Polygon  of  an  infinite  Number  of  Sides, 
each  of  an  infinitely  small  Length,  which  determine 
the  Curvature  of  the  Line  by  the  Angles  they  make 
with  each  other." 

De  l'Hospital's  "  prendre  la  difìférence"  is 
rendered  by  Stone  "  to  find  the  fluxions."  The 
fluxion  oi  xy  is  found  by  taking  the  product  of  x-\-x 
and  J+j,  and  neglecting  xy,  "  because  ij  is  a 
Quantity  infinitely  small,  in  respect  of  the  other 
Terms  yx  and  xj/." 

72.  Further  on  in  Stone's  translation  (p.  73)  we 
read  : 


PRINTED  BOOKS,  ETC,  BEFORE  1734     55 

*'The  infinitely  small  Part  generateci  by  the  con- 
tinuai increasing  or  decreasing  of  the  Fluxion  of  a 
variable  Quantity,  is  called  the  Fluxion  of  the 
Fluxion  of  that  Quantity,  or  second  Fluxion.''  In 
like  manner  he  defines  third  Fluxion  ;  '  '  fluxion  of 
the  second  fluxion"  taking  the  place  of  "  différence 
de  la  différence  seconde." 

In  the  appendix,  containing  Stone's  Inverse 
Method  of  Fluxions,  a  fluent  is  defined  thus  : 

'  '  The  fluent  or  flowing  Quantity  of  a  given 
fluxionary  Expression,  is  that  Quantity  whereof 
the  given  fluxionary  Expression  is  the   Fluxion." 

Remarks 

73.  The  earliest  treatment  of  the  new  analysis 
which  became  current  in  England  was  that  of 
Leibniz.  The  Scotsman  Craig  used  it  for  over 
a  quarter  of  a  century  before  rejecting  it  in  favour 
of  fluxions.  Harris,  Hayes,  and  Stone  drew  their 
inspiration  from  French  writers  who  followed 
Leibniz.  A  hopeless  confusion  arose  in  the  use  of 
the  term  ''fluxion."  Newton  always  took  it  to  be 
a  velocity,  but  many  writers,  including  Newton's 
friends  who  prepared  the  Corniate rcium  Episiolicum, 
simply  said  "fluxion"  instead  of  "  differential,  " 
thus  putting  a  home  label  upon  goods  of  foreign 
manufacture.  A  strict  foUower  of  the  Newton  of 
1704  was  Ditton  ;  fluxions  are  taken  as  infinitesimals 
by  Fatio  de  Duillier,  Cotes  (in  1701),  Harris,  Hayes, 
Raphson,  Stirlìng  (in  17 17),  and  Stone. 

Stone  Comes  out  strongly  with  the  view  that  a 


56  LIMITS    AND   FLUXIONS 

circle  is  a  polygon  of  an  infinite  number  of  sides. 
He  also  uses  the  infinitesimal  triangle.  Hayes  and 
Stone  have  no  hesitation  in  speaking  of  '' fluxions 
of  fluxions,"  and  "  infinitely  little  parts  of  an  in- 
finitely  little  part."  No  writers,  unless  we  except 
Newton  (1704)  and  Ditton,  dispense  with  the  use  of 
infinitely  small  quantities.  The  dropping  of  such 
quantities  from  an  equation  was  usually  permitted 
without  scruple. 

What  an  opportunity  did  this  medley  of  untenable 
philosophical  doctrine  present  to  a  dose  reasoner 
and  skilful  debater  like  Berkeley  !     [See  Addenda, 

p.  289.] 


CHAPTER    III 

BERKELEY'S   ANALYST  '(1734)  ;   CONTROVERSY 
WITH    JURIN    AND   WALTON 

74.  BiSHOP  BERKELEY'S  publication  of  the  Analyst'^ 
is  the  most  spectacular  event  of  the  century  in  the 
history  of  British  mathematics.  The  arguments  in 
the  Analyst  were  so  many  bombs  thrown  into  the 
mathematica!  camp. 

The  views  expressed  in  the  Analyst  are  fore- 
shadowed  in  Berkeley's  Principles  of  Human  Know- 
ledge  (§§  123-134),  published  nearly  a  quarter  of 
a  century  earUer.  The  "  Infidel  mathematician," 
it  is  generally  supposed,  was  Dr.  Halley.  Mathe- 
maticians  complain  of  the  incomprehensibiUty  of 
rehgion,  argues  Berkeley,  ,but  they  do  so  unreason- 
ably,  since  their  own  science  is  incomprehensible. 
"  Our  Sense  is  strained  and  puzzled  with  the 
perception  of  objects  extremely  minute,  even  so 
the  Imagination,  .  .  .  is  very  much  strained  and 
puzzled  to  frame  clear  ideas  of  the  least  particles  of 
time,    or    the    least    increments   generated   therein  : 

^  The  Analyst  :  or,  a  Discourse  addresscd  to  an  Infide l  Mathe- 
matician. Wherein  it  is  exainined  whether  the  Oò/'ect,  Principles,  and 
Inferences  of  the  Modem  Analysis  are  more  dislinctly  concewed,  or 
more  evidently  deduced,  than  relii^otis  Mystenes  and  Points  of  Faith. 
London,  1734. 

57 


58  LIMITS    AND   FLUXIONS 

and  much  more  so  to  comprehend  the  moments,  or 
those  increments  of  the  flowing  quantities  in  staiu 
nascenti,  in  their  very  first  origin  or  beginning  to 
exist,  before  they  become  finite  particles.  And  it 
seems  stili  more  difficult  to  conceive  the  abstracted 
velocities  of  such  nascent  imperfect  entities.  But 
the  velocities  of  the  velocities — the  second,  third, 
fourth,  and  fifth  velocities,  etc. — exceed,  if  1  mistake 
not,  ali  human  understanding  "  {Analyst,  §  4).    .    .    . 

75.  '*In  the  calculus  differentialis  .  .  .  our 
modem  analysts  are  not  content  to  consider  only 
the  differences  of  finite  quantities  :  they  also 
consider  the  differences  of  those  differences,  and 
the  differences  of  the  differences  of  the  first  differ- 
ences :  and  so  on  ad  infiniturn.  That  is,  they 
consider  quantities  infinitely  less  than  the  least 
discernible  quantity  ;  and  others  infinitely  less  than 
those  infinitely  small  ones  ;  and  stili  others  infinitely 
less  than  the  preceding  infinitesimals,  and  so  on 
without  end  or  limit  "  (§  6). 

'j^.  "■  I  proceed  to  consider  the  principles  of  this 
new  analysis.  .  .  .  Suppose  the  product  or  rectangle 
AB  increased  by  continuai  motion  :  and  that  the 
momentaneous  increments  of  the  sides  A  and  B  are 
a  and  b.  When  the  sides  A  and  B  are  deficient,  or 
lesser  by  one-half  of  their  moments,  the  rectangle 
was  À-4^xB-i  ^,  i.e.  h.^-\  a^-l  bK^-\  ab. 
And  as  soon  as  the  sides  A  and  B  are  increased  by 
the  other  two  halves  of  their  moments,  the  rectangle 
becomes  A  +  -|  ^  x  B  +  J  ^  or  AB  +  |  aV>  +  \  bA  +  |  ab. 
From  the  latter  rectangle  subduct  the  former,  and 


BERKELEY' S    ANALYST  (1734)         59 

the  remaining  difìference  will  be  à^-\-bh.  There- 
fore  the  increment  of  the  rectangle  generateci  by 
the  entire  increments  «  and  <^  is  <3:B  +  ^A.  Q.E.D. 
But  it  is  plain  that  the  direct  and  true  method  to 
obtain  the  moment  or  increment  of  the  rectangle 
AB,  is  to  take  the  sides  as  increased  by  their  whole 
increments,  and  so  multiply  them  together,  A+<a:  by 
B  +  <^,  the  product  whereof  AB  +  ^B  +  ^^A +  ^(^  is  the 
augmented  rectangle  ;  whence,  if  we  subduct  AB 
the  remainder  à^ -\- b i^ -\- ab  will  be  the  true  incre- 
ment of  the  rectangle,  .  .  .  and  this  holds  uni- 
versally  by  the  quantities  a  and  b  be  what  they 
will,  big  or  little,  finite  or  infinitesimal,  increments, 
moments,  or  velocities  "  (§  9).  .  .  .  The  point  of 
getting  rid  of  ab  cannot  be  obtained  by  legitimate 
reasoning.  "  .    .    . 

'jj,  "The  points  or  mere  limits  of  nascent  lines 
are  undoubtedly  equal,  as  having  no  more  magnitude 
one  than  another,  a  limit  as  such  being  no  quantity. 
If  by  a  momentum  you  mean  more  than  the  very 
initial  limit,  it  must  be  either  a  finite  quantity 
or  an  infinitesimal.  But  ali  finite  quantities  are 
expressly  excluded  from  the  notion  of  a  momentum. 
Therefore  the  momentum  must  be  an  infinitesimal. 
.  .  .  Por  aught  I  see,  you  can  admit  no  quantity 
as  a  medium  between  a  finite  quantity  and  nothing, 
without  admitting  infinitesimals  "  (§  11). 

78.  Berkeley  next  premises  the  following  lemma, 
which  figures  prominently  in  the  debates  about 
fluxions  : 

"  '  If,  with  a  view  to  demonstrate  any  proposition, 


6o  LIMITS    AND   FLUXIONS 

a  certain  point  is  supposed,  by  virtue  of  which 
certain  other  points  are  attained  ;  and  such  supposed 
point  be  itself  afterwards  destroyed  or  rejected  by 
a  contrary  supposition  ;  in  that  case,  ali  the  other 
points  attained  thereby,  and  consequent  thereupon, 
must  also  be  destroyed  and  rejected,  so  as  from 
thenceforward  to  be  no  more  supposed  or  applied  in 
the  demonstration.'  This  is  so  plain  as  to  need  no 
proof"(§  12). 

79.  Berkeley  examines  now  the  method  of  obtain- 
ing  the  fluxion  oi  x*^  by  writing  x-\-o  in  the  place 
of  ;r,  expanding  by  the  binomial  formula,  writing 
down  the  increments  of  x  and  x"",  which  are  in  the 
ratio  of 

I  to  «;ir«-i  +  ^-^^^?-Z^W-2+  etc, 
2 

or,  when  the  increment  0  is  made  to  vanish,  in  the 
ratio  of  I  to  nx*"'^.      Berkeley  argues  : 

"  But  it  should  seem  that  this  reasoning  is 
not  fair  or  conclusive.  For  when  it  is  said,  let 
the  increments  vanish,  i.e.  let  the  increments  be 
nothing,  or  let  there  be  no  increments,  the  former 
supposition  that  the  increments  were  something,  or 
that  there  were  increments,  is  destroyed,  and  yet  a 
consequence  of  that  supposition,  i.e.  an  expression 
got  by  virtue  thereof,  is  retained.  Which,  by 
the  foregoing  lemma,  is  a  false  way  of  reasoning. 
Certainly  when  we  suppose  the  increments  to  vanish, 
we  must  suppose  their  proportions,  their  expres- 
sions,  and  everything  else  derived  from  the  supposi- 
tion of  their  existence,  to  vanish  with  them  (§  13). 


BERKELEY'S    ANALYST  (1734)         61 

.  .  .  Ali  which  seems  a  most  inconsistent  way  of 
arguing,  and  such  as  would  not  be  allowed  of  in 
Divinity  (§  14).  .  .  .  Nothing  is  plainer  than  that 
no  just  conclusion  can  be  directly  drawn  from 
two  inconsistent  suppositions  (§15).  .  .  .  It  may 
perhaps  be  said  that  [in  the  calculus  differentialis\ 
the  quantity  being  infinitely  diminished  becomes 
nothing,  and  so  nothing  is  rejected.  But,  accord- 
ing  to  the  received  principles,  it  is  evident  that  no 
geometrica!  quantity  can  by  any  division  or  sub- 
division  Yvhatsoever  be  exhausted,  or  reduced  to 
nothing.  Considering  the  various  arts  and  devices 
used  by  the  great  author  of  the  fluxionary  method  ; 
in  how  many  lights  he  placeth  his  fluxions  ;  and  in 
what  different  ways  he  attempts  to  demonstrate  the 
same  point  ;  one  would  be  inclined  to  think,  he  was 
himself  suspicious  of  the  justness  of  his  own  demon- 
strations,  and  that  he  was  not  enough  pleased  with 
any  notion  steadily  to  adhere  to  it  "  (§17).    .    .    . 

80.  ''And  yet  it  should  seem  that,  whatever 
errors  [in  the  calculus  differentialis\  are  admitted 
in  the  premìses,  proportional  errors  ought  to  be 
apprehended  in  the  conclusion,  be  they  finite  or 
infinitesimal  :  and  that  therefore  the  aKpl^eia  of 
geometry  requires  nothing  should  be  neglected  or 
rejected.  In  answer  to  this  you  will  perhaps  say, 
that  the  conclusions  are  accurately  true,  and  that 
therefore  the  principles  and  methods  from  whence 
they  are  derived  must  be  so  too.  But  .  .  .  the 
truth  of  the  conclusion  will  not  prove  either  the  form 
or  the  matter  of  a  syllogism  to  be  true"  (§  19). 


62  LIMITS   AND   FLUXIONS 

8i.  Berkeley  proceeds  to  show  that  correct  results 
are  derived  from  false  principles  by  a  cornpensation 
of  errors^  a  view  advanced  again  later  by  others, 
particularly  by  the  French  critic  L.  N.  M.  Carnet. 
Taking  y'^—px,  Berkeley  says  that  the  subtangent 
is  not  ydx  j  dy  \i  dy  is  the  true  increment  of  y 
corresponding  to  dx  ;  the  accurate  subtangent, 
obtained  by  similar  triangles,  \sydx  /  {dy-{-.z),  where 
z  —  dydy  /  (2j/).  That  is,  if  dy  is  the  true  increment, 
then  in  ydx  I  dy  there  is  an  "error  of  defect.  "  But 
in  ydx  /  dy,  as  used  in  the  differential  calculus,  the 
dy  is  not  its  true  value,  viz.  dy=pdx  /  (2y)  — 
dydy  I  (2y)  (obtained  by  writing  x-\-dx  for  x  and 
y  +  dy  for  j,  in  the  equation  r^=/;i'),  but  its  erroneous 
value,  pdx  I  {2y).  There  is  bere  an  "  error  of 
excess."  "  Therefore  the  two  errors  being  equal 
and  contrary  destroy  each  other  (§  21);  .  .  .  by 
virtue  of  a  twofold  mistake  you  arrive,  though  not 
at  science,  yet  at  truth."  Berkeley  gives  other 
illustrations  of  cases  where  "  one  error  is  redressed 
by  another.  " 

82.  "A  point  may  be  the  limit  of  a  line  :  a  line 
may  be  the  limit  of  a  surface  :  a  moment  may 
terminate  time.  But  how  can  we  conceive  a  velocity 
by  help  of  such  limits  ?  It  necessarily  implies  both 
time  and  space,  and  cannot  be  conceived  without 
them.  And  if  the  velocities  of  nascent  and  evan- 
escent  quantities,  i.e.  abstracted  from  time  and 
space,  may  not  be  comprehended,  how  can  we 
comprehend  and  demonstrate  their  proportions  ;  or 
consider  their  rafiones  prima^  and  ultimceì     For,  to 


BERKELEY'S    ANALYST  (1734)        ^l 

consider  the  proportion  or  ratio  of  things  implies 
that  such  things  have  magnitude  ;  that  such  their 
magnitudes  may  be  measured  "  (§31).    .    .    . 

83.  "  If  it  be  said  that  fluxions  may  be  expounded 
or  expressed  by  finite  lines  proportional  to  them  ; 
which  finite  lines,  as  they  may  be  distinctly  con- 
ceived  and  known  and  reasoned  upon,  so  they  may 
be  substitutpd  for  the  fluxions,  ...  I  answer  that 
if,  in  order  to  arrive  at  chese  finite  Hnes  proportional 
to  the  fluxions,  there  be  certain  steps  made  use  of 
which  are  obscure  and  inconceivable,  be  those 
finite  lines  themselves  ever  so  clearly  conceived,  it 
must  nevertheless  bc  acknowledged  that  your  pro- 
ceeding   is   not   clear   nor   your    method   scientific  " 

(§  34). 

Berkeley  discusses  this  matter  with  reference  to 
a  geometrie  figure,  and  argues  that  "a  point  there- 
fore  is  considered  as  a  triangle,  or  a  triangle  is 
supposed  to  be  formed  in  a  point.  Which  to  con- 
ceive  seems  quite  impossible  "  (§  34).    .    .    . 

84.  "  And  what  are  these  fluxions  ?  The  Veloci- 
ties  of  evanescent  increments.  And  what  are  these 
same  evanescent  increments  ?  They  are  neither 
finite  quantities,  nor  quantities  infinitely  small,  nor 
yet  nothing.  May  we  not  cali  them  the  ghosts  of 
departed  quantities  ?"(§  35).    .    .    . 

"  And  if  the  first  [fluxions]  are  incomprehensible, 
what  shall  we  say  of  the  second  and  third  fluxions, 
etc.?"(§44). 

"  To  the  end  that  you  may  more  clearly  com- 
prehend    the    force    and    design    of    the    foregoing 


64  LIMITS    AND   FLUXIONS 

remarks  .   .   .,  I  shall  subjoin  the  foUowing  Queries  " 

(§  50). 

Then  follow  sixty-seven  queries,  of  which  the 
sixteenth  is  a  good  specimen  :  '' Qu.  16.  Whether 
certain  maxims  do  not  pass  current  among  analysts 
which  are  shocking  to  good  sense  ?  And  whether 
the  common  assumption,  that  a  finite  quantity 
divided  by  nothing  is  infinite,  be  not  of  this 
number  ?  " 

Jurins  First  Reply  to  Berkeley 

85.  A  reply  to  Berkeley 's  Analyst  was  made  by 
the  noted  physician,  James  Jurin,  at  one  time  a 
student  in  Trinity  College,  Cambridge,  who  had 
imbibed  Newtonian  teachings  from  Newton  himself. 
Jurin  wrote  under  the  pseudonym  of  '' Philalethes 
Cantabrigiensis."     The   lettera  is   dated    Aprii   io, 

1734. 

86.  Philalethes  says  that  the  charge  in  the 
Analyst  *'consists  of  three  principal  points  :  (i)  Of 
Infidelity  with  regard  to  the  Christian  Religion. 
(2)  Of  endeavouring  to  make  others  Infidels,  and 
succeeding  in  those  endeavours  by  means  of  tJie 
deference  which  is  paid  to  their  JHdo;i)ient,  as  being 

^  Geovietry  No  Friend  to  Infidelity  :  or,  a  De/enee  of  Sir  Isaac 
Newton  and  the  British  Mathematicians,  In  a  Lettor  to  the  Author  of 
the  Analyst.  Whcrein  it  is  cxainined,  How  far  the  Conduci  of  siich 
Divines  as  intermix  the  Interest  of  Religion  with  their  private  Disfutes 
and  Passions,  and  allow  neither  Learning  nor  Keason  to  those  they 
differfroiiiy  is  of  Hononr  or  Service  to  Christianity,  or  as^reeable  to  the 
Example  of  our  Blessed  Saviour  and  his  Apostles.  By  Philalethes 
Caiitaì)rigiensis.  Ne  Deus  intersit,  nisi  dignus  vindice  nodus  Inciderit. 
London  :  Printed  for  T.  Cooper  at  the  Globe  in  Ivy-L;ine. 
MDCCXXXIV.     Price  is. 


BERKELEY'S    ANALYST  (1734)         65 

presumed  to  be  of  ali  inen  the  greatest  masters  of 
reason.  (3)  Of  error  and  false  reasoning  in  their 
own  science.  " 

87.  The  early  part  of  Jurin's  reply  is  given  to  a 
discussion  of  the  religious  side.  If  there  is  no 
more  certainty  -in  modem  analysis,  argues  Jurin, 
than  in  the  Christian  religion,  this  comparison  brings 
no  honour  to  Christianity  ;  it  is  not  true  that 
mathematicians  are  infidels,  leading  others  to 
infidehty.  If  it  were  true,  this  fact  ought  not  in 
prudence  to  be  published.  Even  if  it  be  shown 
that  the  method  of  fluxions  is  built  upon  false 
principles,  will.  it  follow  that  ali  other  parts  of 
mathematics  rest  on  inaccurate  and  false  reasoning  ? 
Your  attack,  I  surmise,  is  really,  not  so  much  in 
the  interest  of  Christianity,  as  to  demonstrate  your 
superiority  as  a  reasoner,  by  showing  Newton  and 
Barrow,  two  of  the  greatest  mathematicians,  less 
clear  and  just  than  you  are.  But  because  a  mathe- 
matician  "  is  thought  to  reason  well  in  Geometry,  " 
his  '' decisions  against  the  Christian  Religion  "  will 
not  "pass  even  upon  weak  and  vulgar  minds. " 
"  Sir  Isaac  Newton  was  a  greater  Mathematician 
than  any  of  his  contemporaries  in  France,  .  .  .  yet 
I  bave  not  heard  that  the  French  Mathematicians 
are  converted  to  the  Protestant  Religion  by  his 
authority."  Your  objections  against  Newton's 
Fluxions  may  be  "  reduced  under  three  heads  : 
(i)  Obscurity  of  this  doctrine  ;  (2)  False  reasoning 
in  it  by  Sir  Isaac  Newton,  and  implicitly  received 
by  his  followers  ;  (3)  Artifices  and  fallacies  used  by 

5 


66  LIMITS    AND   FLUXIONS 

Sir  Isaac  Newton,  to  makc  this  false  reasoning  pass 
upon  his  followers."  Jurin  continues  :  "  It  must 
be  owned  that  this  doctrine  .  .  .  is  not  without 
difficulties,"  but  "bave  you  not  altered  his  ex- 
pressions  in  such  a  manner,  as  to  mislead  and  con- 
found  your  readers,  instead  of  informing  them," 
thereby  increasing  the  difficulties?  "  Where  do 
you  find  Sir  Isaac  Newton  using  such  expressions 
as  the  velocities  of  the  velocities,  the  second,  thii-d  and 
fourth  velocities,  the  incipient  celerity  of  an  incipient 
celerity,  the  nascent  augnient  of  a  nascent  augvient  ?  " 
As  to  the  ''moment  or  increment  of  the  rectangle 
AB,"  the  mathematicians  take  it  to  be  ^B-f/;A  ; 
you  say  that  the  rigorous  value  is  ^B  +  /;A  +  ^/7. 
*'Do  not  they  know  that  in  estimating  any  finite 
quantity  how  great  soever  .  .  .,  a  globe,  suppose, 
as  big  as  the  earth,  ...  or  even  the  orb  of  the 
fixed  stars  .  .  .,  this  omission  shall  not  cause  them 
to  deviate  from  the  truth  so  much  as  a  single  pin's 
head,  nay  not  the  millionth  part  of  a  pin's  head  ?  " 
The  operations  by  fluxions  are  no  more  objection- 
able  than  those  by  decimai  fractions,  where  we  take 
•33333,  etc,  instead  ofj.  You  say  that  the  Marquis 
de  l'Hospital,  in  his  Analyse  des  infiniment  petits, 
Prop.  2,  having  found  the  fluxion  of  xy  to  be  xdy 
■\-ydx-\-dxdy,  drops  the  dxdy  ''without  the  least 
ceremony.  "  But  does  he  not  especially  require  in 
a  postulate,  "that  a  quantity,  which  is  augmented 
or  diminished  by  another  quantity  infinitely  less 
than  the  first,  may  be  considered  as  if  it  continued 
the  same,  i.e.  had  received  no  such  augmentation  or 


BERKELEY'S    ANALYST  (1734)         ^7 

diminution  ?  "  As  to  Newton,  he  takes  {Principia, 
lib.  ii,  lemma  2,  cas.  i  ;  our  §  17)  initially  {A  —  \a) 
{B-\b)  and  finally  ( A  +  4  «)(I^  +  è  ^),  thereby  de- 
riving  al^-\-bA,  not  as  the  increment  of  AB,  but 
as  the  increment  of  {l\  —  \a){V>  —  \h).  '' .  .  . 
Rigorously  speaking,  the  moment  of  the  rectangle 
AB  is  not,  as  you  suppose,  the  increment  of  the 
rectangle  AB  ;  but  it  is  the  increment  of  the  rect- 
angle A  —  J  <2  X  B  —  J  /^.  "  A  moment  may  be  either 
an  increment  or  a  decrement  ;  you  obtain  the 
increment  aB-\-òA+aò,  the  decrement  of  AB  is 
aB  +  òA  —ab.  Which  of  those  two  will  you  cali  the 
moment  of  AB  ?  "I  apprehend  the  case  will  stand 
thus  :  àB^bA-\-ab-\-aB-\-bA  —  ab  making  twice  the 
moment  of  the  rectangle  AB  ;  it  foUows  that  oB-^bA 
will  make  the  single  moment  of  the  same  rectangle";^ 
the  velocity  which  the  flowing  rectangle  has,  is  its 
velocity  ''neither  before  nor  after  it  becomes  AB, 
but  at  the  very  instant  of  time  that  it  is  AB."  In 
like  manner  with  the  moment  of  the  rectangle. 
Let  me  advise  you  hereafter  to  "first  examine  and 
weigh  every  word  he  [Newton]  uses.  "  Lastly,  1 
must  observe  that  the  moment  of  AB,  namely 
oB-^bA,  and  the  increment  of  the  same  rectangle, 
aB-\-bA-\-ab,  "are  perfectly  and  exactly  equal, 
supposing  a  and  b  to  be  diminished  ad  infiniium.'' 

88.  As  to  your  second  instance  of  false  reason- 
ing,  in  Newton's  hook  on  Quadraiurcs,  apparently 
that  is  "so  truly  Boeotian  a  blunder"  that  I  know 
not  how  "a  Newton  could  be  guilty  of  it. "     You 

^  Jurin,  op.  cit.,  p.  46. 


68  LIMITS    AND   FLUXIONS 

interpret  '' Evanescant  jani  augìiienta  illa'^  (our  §  32), 
as  "let  novv  the  increments  vanish,  i.c,  let  the 
increments  be  nothing,  or  let  there  be  no  incre- 
ments. "  But  '  '  do  not  the  words  ratio  ultima  stare 
US  in  the  face,  and  plainly  teli  us  that  though  there 
is  a  last  proportion  of  evanescent  increments,  yet 
there  can  be  no  proportion  of  increments  which  are 
nothing,  of  increments  which  do  not  exist  ?  "  You 
grossly  misinterpreted  Newton. 

89.  As  to  the  third  head  of  your  objections, 
since  New^ton  did  not  reason  falsely,  "he  had  no 
occasion  to  make  use  of  arts  and  fallacies  to  impose 
upon  his  foUowers."  "  Having  now  .  .  .  driven 
you  entirely  out  of  your  intrenchments  ...  I 
should  Sally  out  and  attack  you  in  your  own.  " 
"But  as  they  seem  rather  designed  for  shew,  than 
use,  .  .  .  to  dazzle  the  imagination  .  .  .  [they] 
will  likewise  immediately  disappear  like  the  Ghost 
of  a  departed  quantity,''  if  you  exorcise  them 
"  with  a  few  words  out  of  the  first  section  of  the 
Principia.'''  You  say  that  the  paradox,  "that 
Mathematicians  should  deduce  true  Propositions 
from  false  Principles  "  is  accounted  for  by  the  fact 
that  one  error  "  is  compensated  by  another  con- 
trary  and  equal  error."  But  the  two  are  no  errors 
at  ali,  as  is  evident  from  the  fact  that  true  results 
foUow  when  only  the  first  operation  is  carried  out, 
so  that  no  compensation  is  possible.  Jurin  argues 
that  the  first  supposed  fallacy,  without  the  second, 
gives  as  the  subtangent  of  y'^=^ax,  the  value 
2x{2y -\- dy) -h- {2y)  \    the     second    supposed    fallacy, 


BERKELETS    ANALYST  (1734)         69 

without  the  first,  gives  2x{2y)-^{2y  +  (fy).  Both 
these  expressions  are  equal  to  2.r,  "  which  is  the 
result  either  of  two  errors,  or  of  none  at  ali."  li 
you  claim  that  2x{2y  +  dy)-^{2y)>  2x,  how  much 
greater  is  it,  supposing  2,r=  1000  miles  ?  Not  as 
much  as  the  thousand-millionth  part  of  an  inch. 
Jurin  ends  vvith  a  discussion  of  Lock  on  abstract 
ideas. 

Wa/ton's  First  Reply  to  Berkeley 

90.  Little  is  known  about  John  Walton.  He 
was  Professor  of  Mathematics  in  Dublin,  and  partici- 
pated  in  this  controversy.  Otherwise,  practically 
nothing  about  him  has  been  handed  down. 

His  reply  to  Berkeley  was  published  in  1735  at 
Uublin.i  Berkeley  attacked  the  method  of  fluxions 
more  particularly  as  given  in  Newton's  earlier 
exposition  ;  Walton  defended  the  theory  on  the 
basis  of  the  later  treatment  as  given  by  Newton 
in  his  Quadratura  Curvaruni  (1704),  and  in  the 
Prmcipia,  Book  II. 

91.  Walton  begins  by  stating  that  inasmuch  as 
the  credulous  may  "  become  infected"  by  Berkeley's 
attack  on  fluxions,  it  seems  necessary  to  give  a 
short  account  of  the  nature  of  fluxions.  "The 
momentaneous   Increments  or  Decrements  of  flow- 

^  A  Vindiiatioìi  of  Sii-  Isaac  Neiuton  x  l'timiples  of  Fluxions, 
agaiusi  the  Oift'idotis  lontaiiied  in  fhe  Analyst.  By  J.  Walton. — 
Siquid  novisti  reclius  isiis,  candidus  imperti  :  bi  non,  his  utere  inecum. 
Hor.  In  the  fuhiess  of  his  Suftìciency  he  shall  he  in  Straits  :  Every 
Hand  of  the  IVickcd  shall  come  upon  him.  y<^/^— Duhlin,  Printed  ; 
and  reprinted  at  London,  and  sold  by  J.  Robcrts  in  Warwick-Lane. 
1735-    [Price  Six  Pence.] 


70  LIMITS    AND   FLUXIONS 

ing  Quantities,  he  [Newton]  elsewhere  calls  by  the 
name  of  Moments,  .  .  :  By  Moments  we  may 
understand  the  nascent  or  evanescent  Elements  or 
Principles  of  finite  Magnitudes,  but  not  Particles 
of  any  determinate  Size,  or  Increments  actually 
generated  ;  for  ali  such  are  Quantities,  themselves 
generated  of  Moments." 

92.  "The  magnitudes  of  the  momentaneous 
Increments  or  Decrements  of  Quantities  are  not 
regarded  in  the  Method  of  Fluxions,  but  their  first 
or  last  Proportions  only  ;  that  is,  the  Proportions 
with  which  they  begin  or  cease  to  exist."  .  .  . 
"The  ultimate  Ratios  with  which  synchronal 
Increments  of  Quantities  vanish,  are  not  the  Ratios 
of  finite  Increments,  but  Limits  which  the  Ratios 
of  the  Increments  attain,  by  having  their  magni- 
tudes infinitely  diminish'd.  .  .  .  There  are  certain 
determinate  Limits  to  which  ali  such  Proportions 
perpetually  tend,  and  approach  nearer  than  by  any 
assignable  Difference,  but  never  attain  before  the 
Quantities  themselves  are  infinitely  diminish'd  ; 
or  'till  the  Instant  they  evanesce  and  become 
nothing."  "The  Pluxions  of  Quantities  are  very 
nearly  as  the  Increments  of  their  Fluents  generated 
in  the  least  equal  Particles  of  Time,"  and  they 
"are  accurately  in  the  first  or  last  Proportions  of 
their  nascent  or  evanescent  Increments."  "The 
Fluxions  of  Quantities  are  only  velocities.  ..." 
Again,  ".  .  .  to  obtain  the  Ratios  of  Fluxions, 
the  corresponding  synchronal  or  isochronal  Incre- 
ments   must    be    lessened    in    infiniiuni.      For    the 


BERKELEY'S    ANALYST  (1734)         71 

Magnitudes  of  synchronal  or  isochronal  Increments 
must  be  infinitely  diminished  and  become  evan- 
escent,  in  order  to  obtain  their  first  or  last  Ratios, 
to  which  Ratios  the  Ratios  of  their  corresponding 
Fluxions  are  equal.  "  The  moment  of  the  rectangle 
AB  is  Ab+Ba,  for  consider  Ab-\-Ba-\-ab  and  Ab  + 
Ba,  "under  a  Constant  Diminution  of  the  Incre- 
ments a  and  b  .  .  .  [they]  constantly  tend  to  an 
Equality  .  .  .  [and]  they  become  equal,  and  their 
Ratio  becomes  a  Ratio  of  Equality.  ..."  Hence 
Ab-\-Ba-\-ab  "  is  not  the  Moment  or  Fluxion  of 
the  Rectangle  AB,  except  in  the  very  Instant 
vvhen  it  begins  or  ceases  to  exist."  Here  fluxions 
appear  to  be  no  longer  velocities  (finite  magnitudes) 
but  moments.  Walton  next  quotes  a  Latin  passage 
from  the  Quadratura  Curvai um.  He  says  that 
Berkeley  seems  ''to  have  been  deceived  by  an 
Opinion  that  there  can  be  no  first  or  last  Ratios 
of  mathematical  Ouantities,"  but  Walton  insists 
that  if  quantities  are  generated  together,  or  if  they 
vanish  together,  they  will  do  so  "under  certain 
Ratios,  which  are  their  first  or  last  Ratios." 
Walton  claims  that  Berkeley's  lemma  "  is  in  no 
Way  pertinent  to  the  Case  for  which  it  was  in- 
tended  "  ;  he  explains  the  Newtonian  process  of 
finding  the  fluxion  of  x*^,  supposing  x  to  increase 
uniformly,  and  points  out  that  this  is  done  without 
rejecting  quantities  "  on  account  of  their  exceeding 
smallness."  Commenting  on  Berkeley's  contention 
that  "no  geometrie  Ouantity,  by  being  infinitely 
diminished,    can     ever     be     exhausted    or     become 


72  LIMITS    AND   FLUXIONS 

nothing,  "  Walton  states  that  the  fluxional  calculus 
assumes  that  '^Ouantities  can  be  generateci  by 
Motion  .  .  .  and  consequently  they  may  also  by 
Motion  be  destroy'd." 

93.  Walton's  Vindication  foUows  Newton's  ex- 
position  closely  ;  Berkeley's  claim  that  Walton 
followed  in  Jurin's  track  and  borrowed  from  him, 
is,  I  believe,  incorrect.  Take  the  vital  question  of 
rejecting  infinitesimals  :  Jurin  claims  that,  being 
so  very  small,  they  do  not  appreciably  affect  the 
result  ;  Walton  takes  the  stand  that  there  is  no 
rejection  whatever  of  infinitesimals.  The  main 
criticism  to  be  passed  on  Walton's  first  essay  con- 
sists,  in  our  judgment,  in  a  failure  to  meet 
Berkeley's  objections  squarely  and  convincingly. 

Beì'kelefs  Reply  to  Jurin  and   Walton 

94.  Jurin's  and  Walton's  articles  were  answered 
by  Berkeley  in  a  publication  entitled,  A  Dcfence  of 
Free-  TJiinking  in  Mathematics.  ^ 

Berkeley  restates  the  purpose  he  liad  in  writing 
the  Analyst '.  "  Novv,  if  it  be  shewn  that  fluxions 
are  really  most  incomprehensible  mysteries,  and 
that  those  who  believe  them  to  be  clear  and  scien- 
tific  do  entertain  an  implicit  faith  in  the  aiithor  of 
that  method  :  will  not  this  furnish  a  fair  argumen- 
tum  ad  hominem  against  men  who  reject  that  very 
thing  in  religion  which  they  admit  in  human  learn- 

^  A  Defence  of  Frec-'J'hitikmi^  in  A/athet/iaiics.  In  Ansivo-  io  a 
Pamphlet  of  Philalethes  Cantabrigiensis.  .  .  .  Also  an  Appendix 
concernine;  Mr.  Walton^ s  Vindication.  .  .  .  By  the  Author  of  "  7'he 
Minute  Philosopher,''   Dublin,    1735. 


BERKELEY'S    ANALYST  (1734)         n 

ing?i  (§  3)  .  .  .  I  say  that  an  infidel,  who  believes 
the  doctrine  of  fluxions,  acts  a  very  inconsistent 
part  in  pretending  to  reject  the  Christian  religion 
— because  he  cannot  believe  what  he  doth  not 
comprehend  "  (§   7).    .    .    . 

^  Berkeley  is  not  the  only  one  who  invoked  the  aid  of  the  Doctrine 
of  Fluxions  in  theological  discussion.  In  a  criticism  {A  Keviexv  of  the 
Fiery  Eniption^  etc,  London,  1752,  p.  128)  of  Bishop  William 
Warburton's  Julian^  concerning  earthquakes  and  fiery  eruptions, 
which,  Warburton  argued,  defeated  Julian's  attempi  to  rebuild  the 
tempie  at  Jerusalem,  it  is  stated  that  a  connection  (needed  in  the 
argument)  was  established  between  the  preservation  of  Christianity 
and  tìie  destruction  of  Judaism  by  the  following  clever  procedure: — 

"The  great  modem  Father  of  the  mathematics  had  invented  a  new 
and  curious  way  of  improving  that  science  by  a  fiction  ;  according  to 
which  quantities  are  supposed  to  be  generated  by  the  continuai  flux  or 
motion  of  others.  In  the  application  of  this  method  it  became  neces- 
sary  to  consider  these  quantities,  sometinìes  in  a  nascent,  and  at  other 
times  in  an  evanescent  state,  by  which  ingenious  contrivance  they 
could  be  made  either  continually  to  tend  to  and  at  last  absolutely  to 
become  nothing,  or  vice  versa,  according  to  the  intention  and  occasions 
of  the  Artist.  Now  by  extending  this  noble  invention  to  the  two 
religions,  it  evidently  appeared,  that,  from  the  time  of  the  first  coming 
of  Christ,  Judaism  entered  into  its  evanescent  state,  as  on  the  other 
band  Christianity  did  into  a  nascent  state,  by  which  means  both  being 
put  into  a  proper  flux,  one  was  seen  continually  decaying,  and  the  other 
continually  improving,  till  at  last  by  the  destruction  of  the  Tempie 
Judaism  actually  vanished  and  became  nothing,  and  the  Christian 
religion  then  bursted  out  a  perfectly  generated  Entity.  .  .  .  As  the 
great  author  of  the  mathematical  method  of  fluxions  had  for  very  good 
reasons  studiously  avoided  giving  any  definition  of  the  precise  magni- 
tude  of  those  moments,  by  whose  help  he  discovers  the  exact  magnitude 
of  the  generated  quantities,  so  our  Author  [Warburton]  by  the  same 
rule  of  application,  and  under  the  influence  of  the  same  authority,  was 
fairly  excused  from  defining  that  precise  degree  of  perfection  and 
imperfection  in  which  the  two  religions  subsisted,  during  the  respective 
evanescent  and  nascent  state  of  each,  by  the  help  of  which  he  discovered 
the  precise  time  when  Judaism  was  perfectly  abolished,  and  Christianity 
perfectly  established.  But  we  may  well  suppose,  that  the  most  alluring 
charm  in  this  extraordinary  piece  of  ingenuity,  was  the  creating  of  a 
new  character  by  it  :  For  questionless  he  may  now  be  justly  stiled  the 
great  founder  and  inventor  of  the  fluxionary  method  of  theology.  .  .  . 
This  fancy  of  a  necessary  connexion  between  the  Temple-edifice,  and 
the  being  of  Christianity,  .  .  .  this  pretended  Christianity  which  is  of 
such  an  unsubstantial  nature,  that  it  must  necessarily  vanish  at  the 
restoration  of  the  Tempie,  can  be  nothing  else  but  a  mere  Ghost^  .  .  . 
evidently  the  Ghost  of  departed  Judaism." 


74  LIMTTS    AND   FLUXIONS 

95.  "  I  bave  said  (and  I  venture  stili  to  say)  that 
a  fluxion  is  incomprehensible  :  that  second,  third, 
and  fourth  fluxions  are  yet  more  incomprehensible  : 
that  it  is  not  possible  to  conceive  a  simple  infini- 
tesimal  :  that  it  is  yet  less  possible  to  conceive  an 
infinitesimal  of  an  infinitesimal,  and  so  onward. 
What  bave  you  to  say  in  answer  to  this  ?  Do  you 
attempt  to  clear  up  the  notion  of  a  fluxion  or  a 
difference  ?     Nothing  like  it  "  (§    17). 

96.  Berkeley  quotes  from  Newton's  Principia 
and  QuadratUì^e  of  Curves,  and  then  asks,  "  Is  it 
not  plain  that  if  a  fluxion  be  a  velocity,  then  the 
fluxion  of  a  fluxion  may,  agreeably  thereunto,  be 
called  the  velocity  of  a  velocity?  In  like  manner, 
if  by  a  fluxion  is  meant  a  nascent  augment,  will  it 
not  then  foUow  that  the  fluxion  of  a  fluxion  or 
second  fluxion  is  the  nascent  augment  of  a  nascent 
augment?"  (§  23). 

97.  "I  had  observed  that  the  great  author  had 
proceeded  illegitimately,  in  obtaining  the  fluxion 
or  moment  of  the  rectangle  of  tvvo  flowing  quan- 
tities.  ...  In  answer  to  this  you  allege  that  the 
error  arising  from  the  omission  .  .  .  is  so  small 
that  it  is  insignificant  (§  24).  .  .  .  If  you  mean 
to  defend  the  reasonableness  and  use  of  approxi- 
mations  ...  I  bave  nothing  to  say.  .  .  .  That 
the  method  of  fluxions  is  supposed  accurate  in 
geometrica!  rigour  is  manifest  to  whoever  considers 
what  the  great  author  writes  about  it  .  .  .  In 
rebus  mathernaticis  errores  guam  minimi  7ion  sunt 
contemnendi''  (§   25  ;  our  §   30). 


BERKELEY'S    ANALYST  (1734)         75 

98.  Berkeley  justifies  his  use  of  the  expression 
''  increment  of  a  rectangle  "  by  quoting  from  Newton 
(our  §  1 7),   "  rectanguli  incrementum  aB  -\-òA." 

'*  You  say  '  you  do  not  consider  AB  as  lying  at 
either  extremity  of  the  moment,  but  as  extended 
to  the  middle  of  it  ;  as  having  acquired  the  one 
half  of  the  moment,  and  as  being  about  to  acquire 
the  other  ;  or,  as  having  lost  one  half  of  it,  and 
being  about  to  lose  the  other.'  Now,  in  the  name 
of  truth,  I  entreat  you  to  teli  what  this  moment 
is,  .  .  .  Is  it  a  finite  quantity,  or  an  infinitesimal, 
or  a  mere  limit,  or  nothing  at  ali  ?  .  .  .  If  you 
take  it  in  either  of  the  two  former  senses,  you  con- 
tradict  Sir  Isaac  Newton.  And,  if  you  take  it  in 
either  of  the  latter,  you  contradict  common  sense  ; 
it  being  plain  that  what  hath  no  magnitude,  or  is 
no  quantity,  cannot  be  divided  "  (§  30). 

"...  You  observe  that  the  moment  of  the 
rectangle  determined  by  Sir  Isaac  Newton,  and  the 
increment  of  the  rectangle  determined  by  me  are 
perfectly  and  exactly  equal,  supposing  a  and  ò  to 
be  diminished  ad  infiniiurn  :  and,  for  proof  of  this, 
you  refer  to  the  first  lemma  of  the  first  section  of 
the  first  hook  of  Sir  Isaac's  Principles.  I  answer 
that  if  a  and  /;  are  real  quantities,  then  ab  is  some- 
thing,  and  consequently  makes  a  real  difference  : 
but  if  they  are  nothing,  then  the  rectangles 
whereof  they  are  coefficients  become  nothing  like- 
wise  :  and  consequently  the  monientuvi  or  ina-e- 
mentuni,  whether  Sir  Isaac's  or  mine,  are  in  that 
case  nothing  at  ali.      As  for  the   above-mentioned 


76  LIMITS    AND   FLUXIONS 

lemma,  .  .  .  however  that  way  of  reasoning  may 
do  in  the  method  of  cxhaustions,  where  quanti ties 
less  than  assignable  are  regarded  as  nothiiig  ;  yet, 
for  a  fluxionist  writing  about  momentums,  to  argue 
that  quantities  must  be  equal  because  they  have  no 
assignable  difference,  seems  the  most  injudicious 
step  that  could  be  taken  ;  .  .  .  for,  it  will  thence 
foUow  that  ali  homogeneous  momentums  are  equal, 
and  consequently  the  velocities,  mutations,  or 
fluxions,  proportional  thereto,  are  ali  likewise 
equal"  (§    32). 

99.  As  regards  Newton's  evanescant  jam  augvienta 
illa{p\xx  §  32),  Berkeley  argues  that  it  means  either 
'Met  the  increments  vanish,"  or  else  "  let  them 
become  infinitely  small,"  but  the  latter  "  is  not  Sir 
Isaac's  sense, "  since  on  the  very  same  page  in  the 
Introduction  to  the  Quadrature  of  Curves  he  says 
that  there  is  no  need  of  considering  infinitely  small 
figures.  Taking  advantage  of  the  fact  that  the 
Newton  of  the  Principia  (1687)  differed  from  the 
Newton  of  the  Quadratura  Curvaruvi  (1704),  Berke- 
ley broke  out  into  the  following  philippic  :  "  Vou 
Sir,  with  the  bright  eyes,  be  pleased  to  teli  me, 
whether  Sir  Isaac's  momentum  be  a  finite  quantity, 
or  an  infinitesimal,  or  a  mere  limit  ?  If  you  say  a 
finite  quantity  ;  be  pleased  to  reconcile  this  with 
what  he  saith  in  the  scholium  of  the  second  lemma 
of  the  first  section  of  the  first  hook  of  his  Principles 
(our  §  12):  Cave  intellii^as  quarititatcs  lìuigniliidiìic 
deierminatas,  sed  cogita  seìuper  diniinucìidas  sinc 
limite.      If  you  say,  an  infinitesimal  ;  reconcile  this 


BERKELEY'S    ANALVST  (1734)         77 

with  what  is  said  in  his  Introduction  to  the  Ouadra- 
tures  (our  §  òZ)'-  Volui  os tendere  quod  in  niethodo 
fluxionuni  non  opus  sii  figuras  infinite  pai'vas  in 
geovietriaìH  introducere.  If  you  should  say,  it  is  a 
mere  limit  ;  be  pleased  to  reconcile  this  with  what 
we  find  in  the  first  case  of  the  second  lemma  in  the 
second  hook  of  his  Principles  (our  §  17):  Ubi  de 
lateribus  A  et  B  deerant  monientorum  dimidia,  etc. — 
where  the  moments  are  supposed  to  be  divided.  I 
should  bc  very  glad  a  person  of  sudi  a  luminous 
intellect  would  be  so  good  as  to  explain  whether 
by  fluxions  we  are  to  understand  the  nascent  or 
evanescent  quantities  themselves,  or  their  motions, 
or  their  velocities,  orsimply  their  proportions  .  .  . 
that  you  would  then  condescend  to  explain  the 
doctrine  of  the  second,  third,  and  fourth  fluxions, 
and  show  it  to  be  consistent  with  common  sense  if 
you  can"  (§  36). 

100.  In  an  appendi x  to  the  Defence  of  Free-Think- 
ing  in  Matheìnatics^  Berkeley  replies  to  Walton, 
stating  that  the  issues  raised  by  him  had  been 
previously  raised  by  "the  other,"  that  he  delivered 
a  technical  discourse  -without  elucidating  anything, 
that  his  sclìolars  have  a  right  to  be  informed  as  to 
the  meaning  of  fluxions  and  should  therefore  ask 
him  "the  foUowing  questions. "  Then  foUow  many 
questions,  of  which  we  give  a  few  : 

"  Let  them  ask  him — Whether  he  can  conceive 
velocity  without  motion,  or  motion  without  ex- 
tension,  or  extension  without  magnitude  ?  .  .  . 
Whether   nothing   be   not   the   product    of   notliing 


78  LIMITS    AND   FLUXIONS 

multiplied  by  something  ;  and,  if  so,  .  .  .  when  ab 
is  nothing,  whether  Ab-\-V>a  be  not  also  nothing  ? 
ì.e.  whether  the  momentum  of  AB  be  not  nothing? 
Let  him  then  be  asked,  what  his  momentums  are 
good  for,  when  they  are  thus  brought  to  nothing  ? 
.  .  .  I  wish  he  were  asked  to  explain  the  differ- 
ence  between  a  magnitude  infinitely  small  and  a 
magnitude  infinitely  diminished.  .  .  .  Let  him  be 
farther  asked,  how  he  dares  to  explain  the  method 
of  Fluxions,  by  the  Ratio  of  magni tudes  infinitely 
diminished,  when  Sir  Issac  Newton  hath  expressly 
excluded  ali  consideration  of  quantities  infinitely 
small?  If  this  able  vindicator  should  say  that 
quantities  infinitely  diminished  are  nothing  at  ali, 
and  consequently  that,  according  to  him,  the  first 
and  last  Ratio's  are  proportions  between  nothings, 
let  him  be  desired  to  make  sense  of  this.  .  .  .  If 
he  should  say  the  ultimate  proportions  are  the 
Ratio's  of  mere  limits,  then  let  him  be  asked  how 
the  limits  of  lines  can  be  proportioned  or  divided  ?  " 

Walton's  Second  Reply  to  Berkeley 

lOi.  In  a  second  reply  ^  to  Berkeley,  Walton 
States  that  in  the  Appendix  to  the  Defence^  Berkeley 
"has  composed  a  Catechism  which  he  recommends 
to  my  Scholars  "  and  which  Walton  quotes.  I  am 
first  to  be  asked,  "  Whether  I  can  conceive  Velocity 
without  Motion,  or  Motion  without  Extension.    .    .    . 

^  J.  Walton,  Catechism  of  the  Author  of  the  Minute  Philosopher 
FuUy  answer'd.  Printed  at  Dublin.  Reprinted  at  London,  and  sold 
by  J.  Roberts,  1735.     It  is  a  pamphlet  of  30  pages. 


BERKELEY'S    ANALYST  (i734)         79 

I  answer,  I  can  conceive  Velocity  and  Motion  in  a 
Point  of  Space  ;  that  is,  without  any  assignable 
Length  or  Extension  described  by  it  .  .  .  for  .  .  . 
if  a  cause  acts  continually  upon  a  given  Thing  .  .  . 
there  must  be  a  continuai  Increase  of  its  Velocity  : 
the  Velocity  cannot  be  the  same  in  any  two 
difìferent  Points,"  as  in  the  case  of  falling  bodies, 
Referring  to  A^  +  B^,  Walton  continues  :  ''I  agree 
with  him  that  nothing  is  the  Product  of  nothing 
multipl'd  by  something  ;  but  must  know  what  he 
means  by  the  vanishing  of  the  Gnomon  ^  and  Sum 
of  the  two  Rectangles  .  .  .  before  I  give  him  a 
direct  Answer.  If  by  vanishing  he  means  that 
they  vanish  and  become  nothmg  as  Areas,  I  grant 
they  do  ;  but  absolutely  deny,  upon  such  an  Evan- 
escence  of  the  Gnomon  and  Sum  of  the  two 
Rectangles  by  the  moving  back  of  the  Sides  of  the 
Gnomon  till  they  come  to  coincìde  with  those  of 
the  Rectangle,  that  nothing  remains.  For  there 
stili  remain  the  moving  Sides,  which  are  now 
become  the  Sides  of  the  Rectangle,  .  .  .  the 
Motion  of  the  Gnomon  is  the  same  with  the  Sum  of 
the  Motions  of  the  Two  Rectangles,  when  they 
evanesce,  and  are  converted  into  the  two  Sides  of 
the  Rectangle  AB.  If  a  point  moves  forward  to 
generate  a  Line,  and  afterwards  the  same  Point 
moves  back  again  to  destroy  the  Line  with  the  very 
same  Degrees  of  Velocity,  in  ali  Parts  of  the  Line 


^  If  a  parallelogram  is  extended  in  length  and  breadth  and  if  the 
originai  parallelogram  be  removed,  the  remaining  figure  is  called  the 
gnomon. 


8o  LIMITS    AND   FLUXIONS 

which  it  had  in  those  Parts  when  moving  forward 
to  generate  it  ;  in  the  Instant  the  Line  vanishes  as 
a  Length  .  .  .  the  generating  point  will  remain, 
together  with  the  Velocity  it  had  at  the  very 
Beginning  of  its  Motion,  And  the  Case  is  the 
very  same  with  respect  to  the  Rectangle  increas- 
ing  by  the  Motion  of  its  Sides."  This  point  is 
elaborated  with  great  fullness.  After  some  illustra- 
tions,  Walton  exclaims  :  *'This  is  a  full  and  clear 
Answer  to  this  part  of  the  catechism,  and  shows 
that  its  Author  has  been  greatly  mistaken  in 
supposing  that  I  explained  the  Doctrine  of  Fluxions 
by  the  Ratio  of  Magnitudes  infinitdy  diminish' d,  or 
by  Proportions  between  nothings.  ...  I  do  not 
wonder  that  this  Author  should  bave  no  clear  Ideas 
or  Conceptions  of  second,  third  or  fourth  Fluxions, 
when  he  has  no  clear  Conceptions  of  the  common 
Principles  of  Motion,  nor  of  the  first  and  last  Ratios 
of  the  isochronal  Increments  of  Quantities  generated 
and  destroyed  by  Motion.  ...  In  order  to  prevent 
my  being  Catechised  any  more  by  this  Author," 
Walton  makes  a  confession  *'of  some  Part  of  my 
Faith  in  Religion." 

Jurin's  Second  Reply  to  Berkeley 

102.  Jurin  brought  out  a  second  publication,^  of 
1 1 2  pages,  which  was  in  reply  to  Berkeley's  Defence 
of  Free-Thinking.  Passing  by  unimportant  pre- 
liminaries,  we  come  to  Jurin's  definitions  of  '*flow- 

'    Tlìi'    Minute    Matheinatìclaii  :    or.     The     FrecThiither    no  Jiist- 
Thinkei.      By  Philaleihes  Cantabiigiensis.     London,  1735. 


BERKELEY'S    ANALYST  (1734)         81 

ing  quantity,  "  "  fluxion  "  ("the  velocity  with  which 
a  flowing  quantity  increases  or  decreases  "),  "  incre- 
ment,"  "  nascent  increment  "  ("an  increment  just 
beginning  to  exist  from  nothing  .  .  .  but  not  yet 
arrived  at  any  assignable  magnitude  how  small  so- 
ever"),  "evanescent  increment"  (similarly  defined). 
He  then  endeavours  to  prove  the  proposition  : 
"The  Fluxions,  or  Velocities  of  flowing  quantities 
.  .  .  are  exactly  in  the  first  proportion  of  the 
nascent  increments,  or  in  the  last  proportion  of  the 
evanescent  increments."  He  insists  that  "the  first 
ratio  of  the  nascent  increments  must  be  the  same, 
whether  the  velocities  be  uniform  or  variable  "  ; 
hence,  "the  nascent  increments  must  be  exactly 
as  the  velocities  with  which  they  begin  to  be 
generated."  In  further  explanation,  Jurin  says  that, 
according  to  Newton,  nascent  increments  are  "  less 
than  any  finite  magnitude,"  "  their  magnitude 
cannot  be  assigned  or  determined,"  "the  proportion 
between  them  .  .  .  being  ali  that  is  requisite  in 
his  Method."  In  further  explanation  of  the  pro- 
portion of  evanescent  increments  he  says,  it  "  is 
not  their  proportion  before  they  vanish,  "  "  nor  is  it 
their  proportion  after  they  bave  vanished,"  "but  it 
is  their  proportion  at  the  instant  that  they  vanish." 
Jurin  then  states  that  Berkeley  has  "  taken  as 
mudi  pains  as  .  .  .  any  man  living,  except  a  late 
Philosopher  of  our  University,  to  make  nonsense 
of  Sir  Isaac  Newton's  principles;  "  There  is  no 
occurrence  in  Newton's  writings  of  "velocity  with- 
out   motion,"    "  motion   without   cxtcnsion,"  which 

6 


82  LIMITS    AND   FLUXIONS 

Berkeley  pretends  to  derive  from  them.  Jurin 
succeeds,  we  think,  in  establishing  the  contention 
that  there  is  no  greater  difficulty  in  explaining  the 
second  or  third  fluxion,  than  there  is  in  explaining 
the  first.  "  The  second  fluxion  is  the  velocity  with 
which  the  first  fluxion  increases."  Jurin  confesses 
that  his  statement  in  his  first  reply  to  Berkeley,  to 
the  effect  that  certain  errors  were  of  "no  significance 
in  practice,"  was  intended  for  popular  consumption, 
for  men  such  as  one  meets  in  London. 

103.  "One  of  them,  indeed,  could  make  nothing 
of  what  I  had  said  about  the  length  of  a  subtangent, 
or  the  magnitude  of  the  orb  of  the  fixed  stars  ;  but 
was  fully  satisfied  by  the  information  given  him  by 
one  of  his  acquaintance  to  the  following  effect.  The 
Author  of  the  Minute  Philosopher  has  found  out  that, 
if  Sir  Isaac  Newton  were  to  measure  the  height  of 
St.  Paul's  Church  by  Fluxions,  he  would  be  out  about 
three  quarters  of  a  hair's  breadth  :  But  yonder  is  one 
Philalethes  at  Cambridge,  who  pretends  that  Sir 
Isaac  would  not  be  out  above  the  tenth  part  of  hair's 
breadth.  Hearing  this,  and  that  two  books  had 
been  written  in  this  controversy,  the  honest  gentle- 
man flew  into  a  great  passion,  and  after  muttering 
something  to  himself  about  some  body's  being  over- 
paid,  he  went  on  making  reflections,  which  I  don't 
care  to  repeat,  as  not  being  much  for  your  honour 
or  mine." 

104.  Jurin  thereupon  takes  up  the  rectangle 
AB.  The  terms  "moment"  and  "  increment  " 
are    involved    in    the   discussion   of   it.       Jurin   de- 


BERKELEY' S    ANALYST  (1734)         83 

clares  :  "I  absolutely  and  fully  agree  with  you 
that  the  incremenium  in  the  conclusion  is  the 
momentum  in  the  Lemma,"  that  "the  momenium 
in  the  Lemma"  is  "the  momentU7n  of  the  rectangle 
AB."  Further,  Jurin  says,  "the  incremenium  in 
the  conclusion  is  manifesti)^  the  excess  of  the 
rectangle  A+|<2xB4-|^,  above  the  rectangle 
K  —  \ay.^  —  \b,  i.  e.  the  increment  of  the  rectangle 
A  —  J<2xB  —  1^.  Therefore  we  are  agreed  that  the 
moment  of  the  rectangle  AB  is  the  increment  of 
the  rectangle  A  — J<a:xB  — |^.  Consequently  you 
were  mistaken  in  supposing  that  the  moment  of  the 
rectangle  AB  was  the  increment  of  the  same  rectangle 
AB.  .  .  .  The  moment  AB  is  neither  the  increment 
nor  the  decrement  of  AB,"  for  if  it  really  was  the 
increment  of  AB,  and  also  its  decrement,  we  would 
have  A<^  +  B^  +  ^^  =  A<^  +  B^  —  ab,  i.  e.  2ab  —  o.  Hence 
the  rectangle  ab  "  is  by  his  own  confession  equal  to 
nothing.  "  Jurin  concludes  that  the  fiuxion  of  AB 
is  not  the  velocity  with  which  the  increment  or 
decrement  of  AB  is  generated,  but  the  "middle 
arithmetical  proportional  between  these  two  velo- 
cities,"  this  being  "in  like  manner  as  I  had 
before  supposed  an  arithmetical  mean  between  the 
increment  and  decrement  of  AB,  which  mean  is  the 
moment  of  AB."  Berkeley  had  considered  four 
definitions  of  a  moment,  that  of  a  finite  quantity, 
or  an  infinitesimal  or  a  mere  limit,  or  nothing  at 
ali  ;  and  he  had  found  each  either  to  contradict 
Newton  or  to  contradict  common  sense.  Jurin  does 
not  accept  "  any  one  of  those  senses."     A  moment, 


84  LIMTTS    AND   FLUXJONS 

says  Jurin,  is  defined  by  Newton  as  ''nascent 
increment,  "  its  magnitude  is  "  utterly  unassignable." 
Jurin  continues  : 

105.  "  You  seem  much  at  a  loss  to  conceive  how 
a  nascent  increment,  a  quantity  just  beginning  to 
exist,  but  not  yet  arrived  to  any  assignable 
magnitude,  can  be  divided  or  distinguished  into 
two  equal  parts.  Now  to  me  there  appears  no 
more  difficulty  in  conceiving  this,  than  in  appre- 
hending  how  any  finite  quantity  is  divided  or  dis- 
tinguished into  halves.  For  nascent  quantities  may 
bear  ali  imaginable  proportions  to  one  another,  as 
well  as  finite  quantities." 

106.  Near  the  dose  Jurin  enters  upon  the  dis- 
cussion  of  Berkeley's  Lemma,  given  in  the  Analyst  : 
"  If  one  supposition  be  made,  and  be  afterwards 
destroy'd  by  a  contrary  supposition,  then  everything 
that  followed  from  the  first  supposition,  is  destroyed 
with  it."  Not  so,  says  Jurin,  when  the  supposition 
and  its  contradiction  are  made  at  different  times. 
''Let  US  imagine  yourself  and  me  to  be  debating 
this  matter,  in  an  open  field,  ...  a  sudden  violent 
rain  falls  .  .  .  we  are  ali  wet  to  the  skin  .  .  .  it 
clears  up  .  .  .  you  endeavour  to  persuade  me  I 
am  not  wet.  The  shower,  you  say,  is  vanished 
and  gone,  and  consequently  your  .  .  .  wetness 
.  .  .  must  bave  vanished  with  it.  "  You  say  that 
your  explanation  of  the  correctness  of  results  as 
due  to  a  compensation  of  crrors,  was  intended  by 
you  to  apply,  not  to  Newton,  but  to  Marquis  de 
l'Hospital  ;    your  statements  werc  such   that  not   I 


BERKELEY' S    ANALYST  (1734)         85 

alone,  but  Mr.  Walton  as  well,  inferred  that  you 
were  charging  Newton  with  committing  doublé 
errors.  The  rest  of  Jurin's  ill-arranged  article  is 
given  either  to  a  renewed  and  fuller  eliicidation  of  bis 
previous  contentions  or  to  poetica!  outbursts.  Sure 
of  the  soundness  of  bis  exposition,  he  exclaims,  '*  I 
meet  with  nothing  in  my  way  but  the  Ghosts  of 
departed  difficulties  and  objections.  " 

Berkeley* s  Second  Reply  to    Walton 

107.  ^dX'ioVi'^Catechisin  .  .  ,  fully  Answeredwdi?, 
followed  by  Berkeley's  Reasons  fo7'  not  replying  to 
Mr.  Walton' s  Full  Answer,  1735.  This  last  reply 
has  been  called  *'a  combination  of  reasoning  and 
sarcasm,"  in  which  "  he  affects  to  treat  bis  opponent 
as  a  disguised  convert.  "  Says  Berkeley  :  "  He 
seems  at  bottom  a  facetious  man,  who,  under  the 
colour  of  an  opponent,  writes  on  my  side  of  the 
question,  and  really  believes  no  more  than  I  do  of 
Sir  Isaac  Newton's  doctrine  about  fluxions,  which 
he  exposes,  contradicts,  and  confutes,  with  great 
skill  and  humour,  under  the  mask  of  a  grave  vindica- 
tion."  Berkeley  objects  to  Walton's  motion  and 
velocity  'Mn  a  point  "  of  space;  ''consider  the 
reasoning  :  The  same  velocity  cannot  be  in  two 
points  of  space  ;  therefore  velocity  can  be  in  a  point 
of  space.  ...  I  can  as  easily  conceive  Mr.  Walton 
should  walk  without  stirring,  as  I  can  bis  idea  of 
motion    without    space.  "^     Newton    calls    absolute 

^  Walton  is  not  consistent  in  bis  use  of  the  terni  "  motion."     In  some 
passages  it  means  translation  ;  in  others  it  means  velocity,  or  else  both 


86  LIMITS    AND   FLUXIONS 

motion  "  a  translation  from  absolute  place  to  absolute 
place, "1  and  relative  motion,  "from  one  relative 
place  to  another.  Mr.  Walton's  is  plainly  neither 
of  these  sorts  of  motion  "  ;  hence,  he  argues  against 
Newton.  "  VVhen  ab  is  nothing,  that  is,  when  a 
and  b  are  nothing,  he  denies  that  A^+  B^  is  nothing. 
This  is  one  of  the  inconsistencies  which  I  leave  the 
reader  to  reconcile."  In  his  Vindication  he  holds 
that,  "  to  obtain  the  last  ratio  of  synchronal  incre- 
ments,  the  magnitude  of  those  increments  must  be 
infinitely  diminished  "  ;  in  his  Catechism  .  .  .  fully 
Answered  "he  chargeth  me  as  greatly  mistaken  in 
supposing  that  he  explained  the  doctrine  of  fluxions 
by  the  ratio  of  magnitudes  infinitely  diminished.  "  ^ 
In  his  Catechism  .  .  .  fully  Answered  "he  tells  us 
that  '  fluxions  are  measured  by  the  first  and  last 
proportion  of  isochronal  increments  generated  or 
destroyed  by  motion.'  A  little  later  he  says,  these 
ratios  subsist  when  the  isochronal  increments  have 
no  magnitude."  Can  "isochronal  increments  sub- 
sist  when   they   have    no    magnitude  "  ?      Berkeley 

translation  and  velocity,  as  when  he  says,  "...  isochronal  increments 
must  be  macie  to  vanish  by  a  Retroversion  of  the  Motion  before  we  can 
obtain  the  Moiions  with  which  they  vanish,  or  begin  to  be  generated  ; 
that  is,  before  we  can  obtain  the  Flujdons  of  the  Quantilies,  the  Name 
given  by  Sir  Isaac  Newton  to  those  Motions."  J.  Walton,  Catechism 
.  .  .  fully  Answered,  pp.  i8,  19. 

^  I.  Newton,  Principia,  Definitions,  Scholium,  def.  viii. 

2  What  Walton  actually  wrote  was,  that  Berkeley  had  been  mistaken 
in  supposing  that  he  explained  fluxions  ''  by  the  Ratios  of  Magnittides 
infinitely  diininish'd,  or  by  Proportions  betiveen  nothingsT  Three 
pages  earlier  Wallon  had  denied  ihat  Newton  and  he  measured  fluxions 
"  by  the  Prop ortions  of  Magnitudes  infinitely  small."  Evidently  Walton 
meant  to  exclude  the  *'  infinitely  small,"  but  used  '  '  magnitudes  infinitely 
diminished"  at  one  time  as  magnitudes  "infinitely  small,"  and  at 
anoiher  time  as  signifying  something  else,  namely,  "increments"  that 
"  vanish." 


BERKELEY' S    ANALYST  (1734)         87 

then  quotes  from  his  own  Analyst:  "  As  it  is  im- 
possible  to  conceive  velocity  without  time  or  space, 
without  either  finite  length  or  finite  duration,  it 
must  seem  above  the  power  of  man  to  comprehend 
even  the  first  fluxions.  "  In  the  endeavour  to 
explain  this  matter,  Walton's  skill  has  been  "  vain 
and  impertinent.  " 

The  Second  Edition  of  Walton's  Second  Reply 
108.  Walton  begins  ^  by  explaining  what  Newton 
means  by  Velocity.  It  is  "the  ratio  of  the 
Quantity  of  Motion  to  the  Ouantity  of  Matter  in 
the  body  "  ;  that  is,  if  V  is  the  velocity,  M  the 
quantity  of  motion,  F  the  force  generating  the 
motion,  D  the  density,  B  the  bulk  or  magnitude, 
W  the  weight,  then  "  V  is  M  /  Q,  and  is  as  F  /  W, 
or  as  F  /  DB,"  for,  '*  the  Quantity  of  Motion  is  the 
Ouantity  of  Matter  and  Velocity  taken  together  ; 
that  is,  M  is  OV  "  (p.  35).  "The  Author  [Berkeley] 
therefore  has  been  grossly  mistaken  in  asserting 
that  Velocity  necessarily  implies  both  Time  and 
Space,  and  cannot  be  conceived  without  them. — 
And  that  there  is  no  Measure  of  Velocity  except 
Time  and  Space."  It  appears  that  "a  body  in 
Motion,  will  bave  a  Velocity  inherent  in  itself 
during  the  Whole  Time  of  its  Motion  :  and  conse- 
quently  there  must  be  a  Velocity  where-ever  the 
Body    is,    exclusive    of   Time    and    Space   ...    its 

^  The  Catechisvi  of  the  Author  of  the  Minute  Philosopher  fully 
answer'd.  The  Second  Edition.  With  an  Appendix,  in  Answer  to  the 
Reasonsfor  not  replying  to  Mr.  Waltoiì s  Tuli  Ans7ver.  By  J.  Walton 
.  .  .  Dublin  :  Printed  by  S.  Powell,  for  William  Smith  at  the  Hercules, 
Bookseller  :  in  Dame-Street,  1735. 


88  LIMITS    AND   FLUXIONS 

[a  point's]  Velocity  will  exist  in  a  Point,  and 
successively  will  exist  in  every  Point  of  Space 
through  which  the  Point  moves  "  (p.  37).  Berkeley 
thinks  that  "  from  the  generateci  Velocity  not 
being  the  same  in  any  two  different  Points  of  the 
described  Space  it  will  not  foUow  that  Velocity 
can  exist  in  a  Point  of  Space.  But  in  this  he  is 
mistaken.  P'or  the  continuai  Action  of  a  Moving 
Force  necessarily  preserves  a  continuai  Velocity  ; 
and  if  the  generated  Velocity  be  not  the  same  in 
any  two  different  Points  of  the  described  Space,  a 
Velocity  must  of  Consequence  exist  in  every  Point 
of  that  Space"  (p.  38).  This  account  of  velocity 
"  is  agreeable  to  Sir  Isaac  Newton's  Notion  of 
Velocity  ;  who  constantly  excludes  described  Space 
from  his  Idea  of  that  Term."  Motion  being 
measured  by  QV,  **the  continuai  translation  of  a 
Body  therefore  into  a  new  Place  is,  .  .  .  an  Effect 
of  this  Tendency  forward  in  the  Body,  and  not  the 
Tendency  itself  ;  consequently  Space  described  is 
an  Effect  of  Velocity,  and  not  Velocity  itself" 
(p.  47).  On  the  question  of  first  and  last  ratios  it 
cannot  be  said  that  Walton  here  throws  new  light. 
He  insists  that  he  explained  fluxions  not  ''by  the 
Ratio  of  Magnitudes  infinitely  diminish'd,  but  by 
the  first  and  last  Ratios  of  Increments  generated  or 
destroyed  in  equal  times  :  that  is,  by  the  Ratios  of 
the  Velocities  with  which  those  Increments  begin 
or  cease  to  exist"  (p.  53).  To  Berkeley 's  charge 
that  Walton  "  supposed.  two  Points  to  exist  at  the 
same  Time  in  one  Point,  and  to  be  moved  different 


BERKELEY'S    ANALYST  (1734)         89 

Ways  without  stirring  from  that  Point,"  Walton 
replies  that  there  is  no  difficulty  in  supposing  two 
points  existing  in  a  given  place  each  having  its  own 
velocity,  but  he  never  said  that  they  can  go 
in  different  directions  "without  stirring  from  the 
Point."  Berkeley,  in  his  remarks  about  the  fourth 
fluxion  of  a  cube,  did  not  observe  ali  the  conditions 
which  he  [Walton]  had  imposed.  "  He  [Berkeley] 
intreats  me  to  explain  whether  Sir  Isaac's  Momentum 
be  a  finite  Quantity,  or  an  Infinitesimal,  or  a  mere 
Limit.  I  teli  him,  that  Sir  Isaac's  Momentum  is  a 
finite  quantity  ;  it  is  a  Product  contained  under  the 
moving  Quantity  and  its  Velocity,  or  under  the 
moving  Quantity  and  first  Ratio  of  that  Space 
described  by  it  in  a  given  Particle  of  Time."  Since 
both  these  factors  are  finite,  the  product  is  finite 
(p.  62).  *'By  Moments  therefore  he  is  not  to 
understand  generated  Increments  of  Fluents,  but 
certain  finite  Products  or  Quantities  of  very  different 
Nature  from  generated  Increments,  expressing  only 
the  Motions  with  which  those  Increments  begin  or 
cease  to  exist  "  (p.  63). 

Rejnarks 
109.    Berkeley's  Analyst  must  be  acknowledged 
to  be  a  very  able  production,  which  marks  a  turning- 
point    in   the   history   of  mathcmatical   thought   in 
Great  Britain. 

His  contention  that  no  geometrical  quantity  can 
be   exhausted   by  division  ^  is    in   consonance  with 

^  See  our  §  79. 


90  LIMITS   AND   FLUXIONS 

the  claim  made  by  Zeno  in  his  "  dichotomy,"  and 
the  claim  that  the  actual  infinite  cannot  be  realised. 
The  modem  reader  may  not  agree  vvith  Berkeley 
on  this  point,  nor  in  the  claim  that  second  or 
third  fluxions  are  more  mysterious  than  the  first 
fluxion.  Nevertheless,  a  reader  of  Berkeley  feels 
that  he  spoke  in  the  Analyst  with  perfect  sincerity. 
Interesting  is  DeMorgan's  comment  :■"•  *'Dishonesty 
must  never  be  insinuated  of  Berkeley.  But  the 
Analyst  was  intentionally  a  publication  involving 
the  principle  of  Dr.  Whately's  argument  against  the 
existence  of  Buonaparte  ;  and  Berkeley  was  strictly 
to  take  what  he  found.  The  Analyst  is  a  tract 
which  could  not  have  been  written  except  by  a 
person  who  knew  how  to  answer  it.  But  it  is 
singular  that  Berkeley,  though  he  makes  his 
fictitious  character  nearly  as  clear  as  afterwards 
did  VVhately,  has  generally  been  treated  as  a 
real  opponent  of  fluxions.  Let  us  hope  that  the 
arch  Archbishop  will  fare  better  than  the  arch 
Bishop." 

I  IO.  Sir  William  Rowan  Hamilton  once  wrote 
De  Morgan  :  "  On  the  whole,  I  think  that  Berkeley 
persuaded  himself  that  he  was  in  earnest  against 
Fluxions,  especially  of  orders  higher  than  the  first, 
as  well  as  against  matter."  To  this  De  Morgan 
replied  :  *'  I  have  no  doubt  Berkeley  knew  that  the 
fluxions  were  sound  enough.  "^ 

^  A.  De  Morgan,  Philosophical  Magazine,  4  S.,  voi.  iv,  1852,  p.  329, 
note. 

^  Life  of  Sir  William  Roivan  Hamilton,  by  R.  P.  Graves,  voi.  iii, 
1889,  p.  581. 


BERKELEY'S    ANALYST  (1734)        91 

II I.  One  is  not  so  easily  convinced  of  the  ability 
and  sincerity  of  Jurin.  That  at  first  he  should 
argue  that  quantities  may  be  dropped  because  small, 
and  afterwards  admit  that  this  argument  was  in- 
tended  for  popular  consumption,  is  not  reassuring.  ^ 
That  he  should  fail  to  see  the  soundness  of  Berke- 
ley's  criticism  of  Newton's  proof  {A-\-\a){^-[-\b) 
-{A-\a){B-\  b)  for  the  increment  of  AB  is 
somewhat  surprising,  even  if  it  must  be  admitted 
that  neither  Walton  nor  any  other  eighteenth- 
century  mathematician  appears  to  bave  seen  and 
admitted  the  defect.  In  this  connection  we  quote 
from  a  letter  which  Hamilton  wrote  De  Morgan 
in  1862  vvhen  Hamilton  was  seeing  his  Elenients  of 
Quaternions  through  the  press  :  ^ 

**When  your  letter  arrived  this  morning,  I  was 
deep  in  Berkeley's  '  Defence  of  Freethinking  in 
Mathematics  '  ;  ...  I  think  there  is  more  than 
mere  plausibility  in  the  Bishop's  criticisms  on  the 
remarks  attached  to  the  Second  Lemma  of  the 
Second  Book  of  the  Principia  ;  and  that  it  is  very 
difficult  to  understand  the  logie  by  which  Newton 
proposes  to  prove,  that  the  motnentum  (as  he  calls 
it)  of  the  rectangle  (or  product)  AB  is  equal  to 
^B  +  ^A,  if  the  momenta  of  the  sides  (or  factors) 
A  and  B  be  denoted  by  a  and  b.  His  mode  of 
getting  rid  of  ab  appeared  to  me  long  ago  (I  must 
confess    it)   to    involve    so   much    of  artifice^   as    to 


^  See  our  §§  97,  102,  103. 

*  Life  of  Stì-  Williaftì  Rowan  Hamilton^  by  R.  P.  Graves,  voi.  iii, 
P.  569. 


92  LIMITS    AND   FLUXTONS 

deserve  to  be  called  sophistical  \  although  I  should 
not  like  to  say  so  publicly.  He  subtracts,  you 
know,  (A-|^)(B-P)  from  (A  +  J  ^)(B  + J /^)  ; 
whereby,  of  course,  ab  disappears  in  the  result. 
But  by  what  righi,  or  what  reason  other  than  to 
give  an  unreal  air  of  simplicity  to  the  calculation, 
does  h.Q  pT-epare  the  products  thus  ?  Might  it  not  be 
argued  similarly  that  the  difference, 

was  the  moment  of  A^  ;  and  is  it  not  a  sufifìcient 
indication  that  the  mode  of  procedure  adopted  is  not 
the  fit  one  for  the  subject,  that  it  quite  masks  the 
notion  of  a  limit  ;  or  rather  has  the  appearance  of 
treating  that  notion  as  foreign  and  irrelevant,  not- 
withstanding  ali  that  had  been  said  so  well  before, 
in  the  First  Section  of  the  First  Book  ?  Newton 
does  not  seem  to  have  cared  for  being  very  consis- 
tent  in  bis  philosophy,  if  he  could  anyway  get  hold 
of  ti-uth,  or  what  he  considered  to  be  such.    ..." 

We  give  also  Hermann  Weissenborn's  objec- 
tion  1  to  Newton's  procedure  of  taking  half  of 
the  increments  a  and  b  ;  with  equal  justice 
one  might  take,  says  he,  (A  +  f  ^)(B  + 1  b)  — 
(A  —  \  a){^  —  \  /;),  and  the  result  would  then  be 
\b-\-^a-\-\ab. 

112.  Walton's  two  (or  three)  articles  do  not 
iieem  to  have  been  read  much.  They  are  seldom 
mentioned.      The   pamphlets   are   now   rare.      Pro- 

^  H.  Weissenborn,  Principien  der  h'òheren  Analysis  iti  ihrer  Ent- 
wickelung  von  Leibniz  bis  auf  Lagrange,  Halle,  1856,  p.  42. 


BERKELETS    ANALYST  (1734)         93 

fessor  G.  A.  Gibson  had  not  seen  them  when  he 
wrote  on  the  Analyst  controversy.^  Walton  seemed 
to  have  a  good  intuitive  grasp  of  fluxions,  but 
lacked  deep  philosophic  insight.  He  showed  him- 
self  inexperienced  in  the  conduct  of  controversies, 
and  did  not  know  how  to  protect  himself  against 
attack  from  a  skilful  adversary. 

113.  It  is  worthy  of  notice  that  Walton  ^  ex- 
preSsed  himself  on  the  nature  of  limits,  by  claiming 
that  the  limit  was  reached.  As  to  the  nature 
of  ''variable  velocity,"  it  is  interesting  to  see  that 
Berkeley  realised  the  difficulty  of  the  concept,  and 
probably  saw  that  there  was  no  variable  velocity  as 
a  physical  fact,  while  Walton  was  compelled  to  take 
refuge  in  less  primitive  mechanical  concepts  in  order 
to  uphold  his  side  of  the  argument.  ^  Unjustiflable 
is  Walton's  identification  of  Newton's  "moment" 
with   "  momentum  "  of  mechanics. 

114.  Berkeley's  Lemma*  was  rejected  by  Jurin 
and  Walton.  We  shall  see  that  it  found  no  recog- 
nition  from  mathematicians  in  England  during  the 
eighteenth  century,  but  was  openly  and  repeatedly 
accepted  as  valid  in  its  application  to  limits,  by 
Woodhouse  at  the  beginning  of  the  nineteenth 
century.  The  Newtonian  derivation  of  the  fluxion 
of  X''  (see  our  §§  32,  41),  accomplished  by  dividing 
both  0  and  {x-\-oy'—x'^  by  the  finite  increment  0, 
and  then  putting  0  equal  to  zero  in  the  quotient,  is 

^  G.  A.  Gibson,  "  The  Analyst  Conlroversy,"  in  Proceedings  of  the 
Edinburgh  Math.  Soc,  voi.  xvii,  1899,  p.  18. 

-  See  our  §  92.  ^  See  our  §  loS.  *  See  our  §§  78,  92,  106. 


94  LIMITS    AND   FLUXIONS 

certainly  open  to  the  logicai  objection  raised  by 
Berkeley.  Eighteenth-century  mathematicians  did 
not  attach  due  importance  to  this  point. 

115.  The  existence  of  infinitesimals  (infinitely 
small  quantities)  was  denied  by  Berkeley,  but,  it 
would  seem,  not  denied  by  Jurin  and  Walton.  AH 
three  finally  abjured  the  philosophy  which  permits 
their  being  dropped  because  so  small.  It  is  well 
known  that  many  mathematicians  of  prominence 
have  believed  in  the  reality  of  such  quantities. 
From  Leibniz  to  Lagrange  ali  Continental  writers 
of  note  used  them.  Lagrange  headed  a  small 
school  that  was  opposed  to  them,  when  he  pub- 
lished  his  Fonctions  analytiques.  There  followed 
a  reaction  against  Lagrange.  De  Morgan  once 
remarked  :  '*Duhamel,  Navier,  Cournot,  are  pure 
infinitesimalists.  Some  of  them  say  an  infinitely 
small  quantity  is  one  which  may  be  made  as  small 
as  you  please.  This  is  an  evasion  ;  but  they  do  not 
mean  that  dx  is  finite.  .  .  .  By-the-way,  Poisson 
was  a  believer  in  the  rea/ùy  of  infinitely  small 
quantities — as   I   am."^ 

"...  For  myself,  I  am  now  fixed  in  the  faith 
of  the  subjective  reality  of  infinitesinial  quantity.  But 
what  an  infinitely  small  quantity  is,  I  know  no 
more  than  I  know  what  a  straight  line  is  ;  but  I 
know  it  is  ;  and  there  I  stop  short.  But  I  do  not 
believe  in  objectively  realised  infinitesimals." 

^  Life  of  Sir  William  Rowan  Hamilton,  by  Robert  P.  Graves, 
voi.  iii,  pp.  572,  580.  Consult  also  De  Morgan's  article,  "'On 
Infinity  ;  and  on  the  Sign  of  Equality,"  in  Trans,  of  the  Cambridge 
Phil.   Society^  voi.  xi,  Cambridge,   187 1  [read  May  16,   1864]. 


BERKELEY'S    ANALYST  (1734)        95 

116.  We  must  not  neglect  to  express  our  appre- 
ciation  of  the  fact  that  Berkeley  withdrew  from  the 
controversy  after  he  had  said  ali  that  he  had  to  say 
on  his  subject.  Some  of  the  debates  that  came 
later  were  almost  interminable,  because  the  par- 
ticipants  continued  writing  even  after  they  had 
nothing  more  to  say. 


CHAPTER  IV 

JURIN'S  CONTROVERSY  WITH  ROBINS  AND 
PEMBERTON 

Robins's   '  '  Discourse^  "  and  Review  of  it 

117.  Benjamin  Robins  was  a  native  of  Bath  and 
a  self  -  educateci  mathematician  of  considerable 
reputation. 

The  debate  carried  on  by  Bishop  Berkeley  with 
Jurin  and  VValton  induced  Benjamin  Robins  to 
issue  a  publication,  entitled,  A  Discourse  Concerning 
the  Nature  and  Certamty  of  Sir  Isaac  Newton's 
Methods  of  FluxionSy  and  of  Prime  and  Ultimate 
Ratios,  1735.^  Evidently  Robins  felt  that  Berkeley 's 
attacks  should  be  met,  and  that  Jurin  was  not  the 
man  to  defend  Newton  satisfactorily.  Robins  was 
a  man  of  mathematica!  power  ;  his  exposition  is 
regarded  by  Professor  G.  A.  Gibson  as  very  able, 
and  far  superior  to  that  of  Jurin.  ^  Without  naming 
either  Berkeley  or  Jurin,  and  without  referring  to 
their  articles,  Robins  proceeds  to  his  task.  The 
whole   foundation    of    the    doctrine    of   fluxions    is 

'  This  paper  is  repuhlished,  along  with  subsequent  articles  on  the 
same  subject,  in  the  Matheniatical  Tracts  of  the  late  Benjamin  Robins, 
E^q.  ...  in  two  volumes,  edited  by  James  Wilson,  M.D.  London, 
1761,  voi.  ii,  pp.   1-77. 

-  G.  A.  Gibson,  Ice.  cit.,  pp.  22-25. 

96 


JURIN  V.    ROBIN S    AND   PEMBERTON  97 

based  by  Robins  upon  the  following  two  defini- 
tions  and  certain  general  propositions  annexed  to 
them  : 

1.  Definition  :  *'  .  .  .  we  .  .  .  define  an  ulti- 
mate magnitude  to  be  the  limit,  to  which  a  varying 
magnitude  can  approach  within  any  degree  of 
nearness  whatever,  though  it  can  never  be  made 
absolutely  equal  to  it." 

Here  for  the  first  time  is  the  stand  taken  openly, 
clearly,  explicitly,  that  a  variable  (say  the  peri- 
meter  of  a  polygon  inscribed  in  a  circle)  can  never 
reach  its  limit  (the  circumference).  The  gain  from 
the  standpoint  of  debating  is  very  great  ;  a  regular 
inscribed  polygon  vvhose  sides  are  steadily  doubling 
at  set  intervals  of  time,  say,  every  half  second, 
presents  a  picture  to  the  imagination  which  is 
comparatively  simple.  The  hopeless  attempt  of 
imagining  the  limit  as  reached  need  not  be  made. 
But  this  great  gain  is  made  at  the  expense  of 
generality.  Robins  descends  to  a  very  special  type 
of  variation  which  is  not  the  variation  encountered 
in  ordinary  mechanics  ;  it  is  an  exceedingly  artificial 
variation.  According  to  Robins's  definition,  Achilles 
never  caught  the  tortoise.  It  would  not  be  difficult 
to  assume  a  time  rate  in  the  doubling  of  the  sides 
of  a  polygon  inscribed  in  a  circle,  so  that  the  cir- 
cumference is  7'eached.  Thus,  let  the  first  doubling 
of  the  number  of  sides  take  place  in  i  second,  the 
second  doubling  in  \  a  second,  the  third  m  \  2. 
second,  and  so  on.  It  is  easy  to  see  that  under 
this  mode   of  variation  the  polygons   do   reach  the 

7 


98  LIMITS    AND   FLUXIONS 

limita  the  circumference.  The  process  here  tran- 
scends  our  power  of  imagination,  but  lies  withiii  the 
limits  of  reason.  We  are  dwelling  upon  this  point 
because  of  its  extreme  importance  in  the  history  of 
the  theory  of  Hmits. 

II 8.  Robins  constructs  upon  his  first  definition 
the  theorem,  "  that,  when  varying  magnitudes  keep 
constantly  the  same  proportion  to  each  other,  their 
ultimate  magnitudes  are  in  the  same  proportion." 
As  a  coroUary  of  this  he  states  **that  the  ultimate 
magnitudes  of  the  same  or  equal  varying  magni- 
tudes are  equal." 

II.  Definition:  "  If  there  be  tvvo  quantities, 
that  are  (one  or  both)  continually  varying,  either 
by  being  continually  augmented,  or  continually 
diminished  ;  though  the  proportion,  they  bear  to 
each  other,  should  by  this  means  perpetually  vary, 
but  in  sudi  a  manner,  that  it  constantly  approaches 
nearer  and  nearer  to  some  determined  proportion, 
and  can  also  be  brought  at  length  in  its  approach 
nearer  to  this  determined  proportion  than  to  any 
other,  that  can  be  assigned,  but  can  never  pass  it  : 
this  determined  proportion  is  then  called  the  ulti- 
mate proportion,  or  the  ultimate  ratio  of  those 
varying  quantities." 

Theorem:  **To  this  definition  of  the  sense,  in 
which  the  term  ultimate  ratio,  or  ultimate  pro- 
portion is  to  be  understood,  we  must  subjoin  the 
following  proposition  :  That  ali  the  ultimate  ratios 
of  the  same  varying  ratio  are  the  same  with  each 
other." 


JURIN  V.    ROBINS   AND   PEMBERTON   99 

119.  Robins  remarks  thereupon  that  attempts  at 
the  exposition  of  this  method,  so  far  as  it  depends 
upon  his  first  definition,  vvere  made  by  Lucas 
Valerius  in  a  treatise  on  the  centre  of  gravity,  and 
by  Andrew  Tacquet  in  a  treatise  on  the  cylindrical 
and  annular  solids  ;  but  the  development  involving 
his  second  definition  was  first  made  by  Newton. 
There  are  a  number  of  writers,  not  mentioned  by 
Robins,  who  might  be  cited  as  forerunners  in  the 
theory  of  limits  ;  such,  for  instance,  as  Gregory 
St.  Vincent  and  Stevin. 

Newton's  definition  of  momenta  as  the  momentane- 
ous  increments  or  decrements  of  varying  quantities, 
'*  may  possibly  be  thought  obscure.  "  Robins  eluci- 
dates  thus  :  "In  determining  the  ultimate  ratios 
between  the  contemporaneous  differences  of  quanti- 
ties, it  is  often  previously  required  to  consider  each 
of  these  differences  apart,  in  order  to  discover,  how 
much  of  those  differences  is  necessary  for  expressing 
that  ultimate  ratio"  (§  154).  For  instance,  Kb-\-V>a 
only,  and  not  the  whole  increment  Aò-}-Ba-\-aòf  is 
called  the  momentum  of  the  rectangle  under  A,B. 

120.  Of  this  Discourse,  a  long  account  of  twenty- 
six  pages,  written  by  Robins  himself,  although  his 
name  does  not  appear,^  was  given  in  The  Present 
State  of  the  Republick  of  Letters,  London,  October, 
1735,  in  which  it  is  stated  that  Robins  wrote  his 
Discourse  with  the  view  of  removing  the  doubts 
which    had    lately   arisen    concerning    fluxions    and 

^  This  account  is  republished  in  the  Maihematical  Tracis  of  the  late 
Benjamin  Robins,  edited  by  James  Wilson,  London,  1761,  voi.  ii,  p.  78. 


loo  LIMITS    AND   FLUXIONS 

prime    and  ultimate  ratios  ;  that    Robins    carefully 
distinguished  both  these  methods  from  the  method 
of  indivisibles    and    also  from   each    other.      After 
an  historical   excursion  viewing   the  works   of  the 
ancients,  of  Cavalieri  and  Wallis,  the  introduction 
by  Newton  of  the  concept  of  motion  is  taken  up. 
"  If  the  proportion  between  the  celerìty  of  increase 
of  two  magnitudes  produced  together  is  in  ali  parts 
known,  "  then  ''  the  relation  between  the  magnitudes 
themselves  must  from  thence  be  discoverable."    This 
is  the  basis  for  fluxions.      The    "method  of  prime 
and    ultimate    ratios    proceeds    entirely    upon    the 
consideration  of  the  increments  produced."     By  it 
Newton  avoids    "the  length  of  the  ancient  demon- 
strations  by  exhaustions,  "  on  which,   according  to 
Robins,   the  method  of  fluctions  rests.      "Newton 
did  not  mean,  that  any  point  of  time  was  assign- 
able,     wherein    these    varying    magnitudes    would 
become    actually    equal,    or    the    ratios    really    the 
same  ;  but  only  that  no  difìference  whatever  could 
be  named,  which  they  should  not  pass."     Newton's 
term  momentum  is  used  simply  for  greater  brevity, 
hence  need  not  be  considered.      Newton's  descrip- 
tion    is    capable    of    an    interpretation    too    much 
resembling  the  language  of  indivisibles — in  fact,  he 
sometimes  did  use  indivisibles  at  first  ;  Robins  has 
freed  the  doctrine  from  this  imputation  in  a  manner 
that    "shall    agree    to    the   general    sense    of    his 
[Newton's]  description.  " 


JURIN  V.   ROBINS  AND  PEMBERTON    loi 

Jurìn^s  Review  of  his  own  Lettei's  to  Berkeley 

121.  In  the  November,  1735,  number  of  the 
Republick  of  Letters,  Philalethes  Cantabrigiensis 
(Jurin)  appears  with  an  article,  Considerations  upon 
some  passages  contained  in  two  Letters  to  the  Author 
of  the  Analyst.  The  two  letters  in  question  are  the 
two  repHes  Jurin  himself  had  made  to  Berkeley. 
The  article  is  really  a  reply  to  Robins,  though 
Robins's  name  is  not  mentioned.  Jurin  claims  to 
have  adhered  strictly  to  Newton's  language  ;  some 
other  defenders  of  Newton,  says  he,  are  guilty  of 
departing  from  it.  Their  objections  to  his  own 
defence  are  threefold  : 

"  I.  My  explication  of  Lemma  i,  Lib.  I,  Princip." 
See  our  §§  4,  6,  8. 

'*  IL  The  sense  of  the  Scholium  ad  Sect.  i, 
Libr.  I,  Princ,  particularly  as  to, 

*'  I.   The  doctrine  of  Limits,  2.      The  meaning 
of  the  term  evanescent,  or  vanishing.  " 
See  our  §§  10-15. 
"III.    The  sense  of  Z67;2w<2:  2,  Lib.    \\,  Princip." 
See  our  §§  16-19. 

122.  As  to  the  first  objection,  Jurin  insists  that 
Newton's  v^oràsfitcnt  ultimo  cequales  mean  that  the 
quantities  'Mo  at  last  become  actually,  perfectly, 
and  absolutely  equal."  He  takes  the  hands  of  a 
clock  between  11  and  12.  The  arcs  traced  by  the 
hands  ''  i.  Constantly  tend  to  equality,  2.  During 
an  hour,  3.  And  will  come  nearer  to  one  another 
than    to    have  any  given  difference,   4.  Before  the 


I02  LIMITS    AND   FLUXIONS 

end  of  the  hour  ;  .  .  .  at  the  end  of  the  hour,  the 
two  quantities  must  become  equal.  "  Further,  "  by 
taking  the  consideration  of  a  finite  time,  Sir  Isaac 
Newton  is  able  to  assign  a  point  of  time,  at  which 
he  can  demonstrate  the  quantities  to  be  actually 
equal."  Consider,  says  Jurin,  the  ordinate  to  a 
point  of  a  hyperbola  and  that  ordinate  continued 
to  the  asymptote  :  they  do  not  become  equal  in 
a  finite  time;  Newton's  Lemma  is,  **with  great 
judgment,  so  worded  on  purpose,  as  necessarily  to 
exclude  this  and  such  like  cases.  "  Thus  Newton's 
inscribed  and  circumscribed  rectangles  of  Principia^ 
Lib.  I,  Sec.  I,  Lemma  2  (fig.  i  in  our  §  9),  were 
thought  by  Nieuwentiit  and  others  never  to  be 
capable  of  coincidence  with  the  curve  (say  the 
quadrant  of  a  circle)  ;  but  Jurin  assujues  the  varia- 
tion  to  be  of  such  a  nature  that  the  limit 
is  actually  reached,  as  demanded  by  Newton's 
Lemma.  For,  suppose  a  point  to  move  on  the 
horizontal  radius  from  the  circumference  to  the 
centre  A  in  one  hour  ;  suppose  also  that,  when 
this  moving  point  is  at  B  on  that  radius,  there 
be  two  rectangles  described  upon  AB  (one  in- 
scribed, the  other  circumscribed),  and  that  upon 
every  other  part  of  the  horizontal  radius  that  is 
equal  to  AB,  namely  the  parts  BC,  CD,  DE,  taken 
in  order,  rectangles  be  similarly  erected  ''at  the 
same  point  of  time,"  then  as  the  moving  point 
nears  the  centre,  the  rectangles  diminish  in  size 
and  increase  in  number,  and  they  must  together 
become   equal   to   the   quadrant  at    the  end   of  the 


JURIN  V.   ROBINS  AND  PEMBERTON    103 

hour.  Jurin  points  out  that  he  has  introduced 
bere  ali  the  suppositions  of  Newton's  first  Lemma^ 
namely  that,  (i)  the  two  figures  tend  constantly  to 
equality,  (2)  during  one  hour,  i.e.  a  finite  time,  (3) 
and  come  nearer  to  one  another  than  to  any  gìven 
difìference,  (4)  before  the  end  of  the  hour,  i.e, 
before  the  end  of  a  finite  time.      Jurin  continues  : 

"  If  any  man  shall  say,  that  a  right-angled 
figure,  inscribed  in  a  curvilineal  one,  can  never  be 
equal  to  that  curvilineal  figure  ;  much  less  to 
another  right-lined  figure,  circumscribed  about  the 
curve  ;  I  agree  with  him.  I  am  ready  to  own  that, 
during  the  hour,  these  figures  are  one  inscribed, 
and  the  other  circumscribed  ;  that  neither  of  them 
is  equal  to  the  curvilineal  figure,  much  less  one  to 
another.  But  then,  on  the  other  band,  it  must  be 
granted  me,  that,  at  the  instant  the  hour  expires, 
there  is  no  longer  any  inscribed  or  circumscribed 
figure  ;  but  each  of  them  coincides  with  the  curvi- 
lineal figure,  which  is  the  limit,  the  limes  curvi- 
lìneus,   at  which  they  then  arrive.  " 

123.  Jurin  thereupon  proceeds  to  Lemma  7  of 
Book  I,  Section  i  in  Newton's  Principia,  which,  he 
says,  requires  additional  consideration.  It  relates 
to  fig.  zj,  where  ACB  is  any  are  and  "the  points  A 
and  B  approach  one  another  and  meet."  Lemma  7, 
in  Andrew  Motte's  translation,  reads  as  follows  : — 

''The  same  things  being  supposed  ;  I  say,  that 
the  ultimate  ratio  of  the  are,  chord,  and  tangent, 
any  one  to  any  other,  is  the  ratio  of  equality." 

Jurin  says  that  bere  the  chord  AB,  the  arch  ACB, 


104 


LIMITS   AND  FLUXIONS 


FlG. 


and  the  tangent  AD  come  to  vanish  when  B  reaches 
A,  and  their  last  ratio  is  unity.  Newton  "  directs 
Olir  imagination,  not  to  these  vanishing  quantities 
themselves,  but  to  others  which  are  proportional 
to  them,  and  always  preserve  a 
finite  magnitude,"  such  as  A<^, 
the  arch  Kcb^  Ad.  Since  at  the 
instant  when  A  and  B  coincide, 
''the  angle  BAD,  or  bAd,  will 
vanish  ;  it  is  easy  to  conceive 
that,  .  .  .  the  chord  Ab  must 
coincide  with  the  tangent  Ady 
.  .  .  consequently,  AB,  AD 
must  likewise,  at  the  same  instant  of  time,  arrive 
at  the  same  proportion  of  a  perfect  equality.  " 

124.  Proceeding  to  the  last  Scholium  in  Book  I, 
Section  I  of  the  Principia,  Jurin  starts  by  defining 
the  word  liniit.  "  I  apprehend  therefore  that,  by 
the  limit  of  a  variable  quantity,  is  meant  some 
determinate  quantity,  to  which  the  variable  quantity 
is  supposed  continually  to  approach,  and  to  come 
nearer  to  it  than  to  have  any  given  difiference,  but 
never  to  go  beyond  it.  And  by  the  limit  of  a 
variable  ratio,  is  meant  some  determinate  ratio,  to 
which  the  variable  ratio  is  supposed  continually  to 
approach,  and  to  come  nearer  to  it  than  to  have  any 
given  difference,  but  never  to  go  beyond  it.  By 
arriving  at  a  limit  I  understand  Sir  Isaac  Newton 
to  mean,  that  the  variable  quantity,  or  ratio, 
becomes  absolutely  equal  to  the  determinate  quan- 
tity, or  ratio,  to  which  it  is  supposed  to  tend." 


fURIN  V.  ROBINS  AND  PEMBERTON    105 

VVith  unusual  lucidity,  for  that  period,  Jurin  says 
Oli  the  subject  of  limits  :  *  '  Now  whether  a  quantity, 
or  ratio,  shall  arrive  at  its  limit,  or  shall  not  arrive 
at  it,  depends  entirely  upon  the  supposition  we 
make  of  the  time,  during  which  the  quantity,  or 
ratio,  is  conceived  constantly  to  tend  or  approach 
towards  its  limit."  If  we  assume  the  approach  to 
be  made  in  a  finite  time,  the  limit  is  reached,  other- 
wise  it  is  not  reached.  Of  a  variable  which  "can 
never  attingere  limitem  "  Newton  gìves  one  illustra- 
tion  at  the  end  of  the  Scholium  :  that  of  two 
quantities  having  at  first  a  common  difference  and 
increasing  together  by  equal  additions,  ad  infinitum. 
Since  they  can  never  be  really  and  in  fact  increased 
ad  infinitum,  says  Jurin,  their  ratio  cannot  arrive  at 
its  limit.  What  Newton  wanted  to  meet  was  the 
objection,  "that  if  the  last  ratio's  of  evanescent 
quantities  could  be  assigned,  the  last  magnitudes 
of  those  quantities  might  likewise  be  assigned." 
Newton  says  No,  "for  those  last  ratio's,  wìth  which 
the  quantities  vanish,  strictly  speaking,  are  not  the 
ratio's  of  the  last  quantities  ...  but  they  are  the 
limits"  which  those  ratios  can  never  "arrive  at," 
"before  the  quantities  are  dÀvcì\m^\i^à.  ad  infinitum.'' 
As  to  the  sense  in  which  Newton  uses  the  word 
evanescent  or  vanishing,  in  the  Scholium  under 
consideration,  Jurin  inclines  to  the  view  that  '*both 
imply  one  single  instant,  or  point  of  time." 

125.  In  the  Principia,  Book  II,  Section  2, 
Lemma  2  (our  §§  16-19),  Newton  defines  moment 
as  "  a  momentaneous  increment,  or  decrement,  of  a 


io6  LIMITS    AND   FLUXIONS 

flowing  quantity,  proportional  to  the  velocity  of  the 
flowing  quantities."  According  to  Jurin,  Newton 
puts  a,  b,  e  to  signify  either  the  moments,  or  the 
velocities,  of  the  flowing  quantities  A,  B,  C. 
Leibniz  looks  upon  them  as  differences.  Newton, 
says  Jurin,  never  used  indivisibles,  and  his  method 
to  find  the  differences  of  variable  quantities  is  not 
*'rigorously  geometrical,"  but  is  more  rigorous 
than  the  treatment  given  by  Leibniz. 

Robins's  Rejoinder 
126.  Robins  replied  in  the  Republick  of  Letters 
for  December,  1735,  in  a  Rcview  of  some  of  the 
Principal  Objections  that  have  been  made  to  the 
Doctrine  of  Fluxions  and  Ultimate  Propoj'tions  ;  with 
some  Remarks  on  the  different  Methods  that  have  been 
taken  to  obviate  them.  Robins  does  not  here  men- 
tion  Philalethes  any  more  than  the  latter  referred 
directly  to  Robins.  The  objections  to  fluxions, 
says  Robins,  are  levelled  at  Newton's  expression, 
fluxiones  sunt  in  ultima  ratione  decrementoi-um  evan- 
escentium  vel  prima  nascentium.  "  Which  being 
usually  thus  translated,  that  fluxions  are  in  the 
ultimate  ratio  of  the  evanescent  decrements,  or  in 
the  first  ratio  of  the  nascent  augments,  it  has  from 
hence  been  ask'd,  what  these  nascent  or  evanescent 
augments  are  ?  "  There  are  difficulties  of  interpre- 
tation,  whether  the  augments  have  quantity  or 
have  not.  One  way  out  of  this  difiiculty  which 
has  been  pointed  out,  is  to  say  :  "the  limit  of  the 
proportion  that  the  decrements  bear  to  each  other 


JURIN  V.  ROBINS  AND  PEMBERTON    107 

as   they   diminish,    is    the    true    proportion    of   the 
fluxions"(p.    438).      Here  a  new  difficulty  arises  : 
Does  the  varying   ratio  reach  its  limit   **actually, 
perfectly,  and  absolutely,"  or  does  it  not  ?     On  the 
understanding  that  it  does  not  reach  the  Hmit,   ''  ali 
that  has  at   any   time   been   demonstrated   by   the 
ancient  method  of  exhaustions  may  be  most  easily 
and  elegantly  deduced."     Rigour  of  demonstration 
does   not   require   ultimate   coincidence.       *'Coinci- 
dence  of  the  variable  quantity  and  its  limit,  could 
it  be  always  prov'd,  would  yet  bring  no  addition  to 
the    accuracy   of   these    demonstrations  "   (p.    441). 
Hence,    *'why    to    the    naturai    difficulty   of   these 
subjects  should  the  obscurity  of  so  strained  a  con- 
ception  be  added  ?  "     Is  this  view  a  correct  inter- 
pretation  of  Newton  ?     A  literal  translation  of  bis 
Lemma    i,   Section    i,   Book    I,  Principia    (see   our 
§§    4,    6,    8),   is  :    "  Ouantities,    and    the    ratios    of 
quantities.    that  during  any  finite  time   constantly 
approach   each  other,    and   before   the   end   of  that 
time   approach   nearer    than    any   given   difference, 
are    ultimately   equal.  "      What    is    the   meaning   of 
*' given  difference"?      If  it   be  a   "difference   first 
assigned"  according  to  which  the  degree  of  approach 
of  these   quantities  may  be  afterwards   regulated  ; 
then  .    .    .    ratios,    and    their    limits,    tho'   they   do 
never   actually  coincide,   will   come  within  the  de- 
scription  of  this  Leinuia  ;  since  the  difference  being 
once  assign'd,  the  approach  of  these  quantities  may 
be  so  accelerated,  that  in  less  than  any  given  time 
the  variable  quantity,  and  its  limit,  shall  differ  by 


io8  LIMirS    AND   FLUXIONS 

less  than  the  assign'd  difìference.  "  Mere  Robins 
expresses  the  idea  that  the  rapidity  of  approach 
toward  the  limit  can  be  arbitrarily  altered,  yet  he 
does  not  apparently  perceive — certainly  he  does  not 
admit — that  this  rapidity  may  be  altered  in  such  a 
way  that  the  variable  actually  does  reach  its  limit. 
On  the  contrary,  he  maintains  that  *'where  the 
approach  is  determin'd  by  a  subdivision  into  parts," 
**it  is  obvious,  that  no  coincidence  can  ever  be 
obtain'd."  A  coincidence  such  as  Philalethes 
explains  in  the  case  of  rectangles  circumscribed 
and  inscribed  in  a  curve,  if  it  could  take  place,  is 
not  a  coincidence  such  as  Newton  intended,  for 
Newton  did  not  in  this  instance  use  motion,  but 
continuai  subdivision.  Robins  tries  to  establish 
his  view  of  the  matter  by  giving  an  instance  of 
erroneous  results  being  deduced  by  letting  the 
variable  reach  its  limit.  He  takes  a  hyperbola 
and  revolves  its  principal  axis  in  the  piane  of  the 
curve,  around  the  point  of  intersection  of  this  axis 
and  an  asymptote,  until  the  two  lines  coincide. 
At  the  end  '*the  hyperbola  coincides  with  the 
asymptote,"  which  is  "absurd."  As  a  matter  of 
fact  there  is  no  absurdity.  In  lAx^  —  a?'y'^  —  a^U^ ^ 
the  slope  of  the  asymptote  is  m  =  b  j  a.  Robins's 
process  amounts  to  making  in  =  o,  which  gives  a 
real  locus  when  b  =  o,  namely  the  \oc\xs  y^  =  o.  The 
only  objection  lies  in  stili  calling  the  final  curve  a 
"  hyperbola." 


JURIN  V.   ROBINS   AND  PEMBERTON    109 

The  Debate  Continued 

1 27.  Robins's  article  was  foUowed  in  the  January, 
1736,  number  of  the  Republick  of  Letters  by 
Philalethes's  Considerations  occasioned  by  a  Paper  in 
the  last  Republick  of  Letters^  concerning  some  late 
Objections  against  the  Do  et  r ine  of  Fluxioìis^  and  the 
different  Methods  that  have  beeii  taken  to  obviate 
them,  Jurin  denies  having  said  that  there  was 
an  **  intermediate  state"  between  augments  being 
**any  real  quantity"  and  being  "actually  vanished"; 
he  says  he  gave  Newton's  declaration  that  "  their 
magnitude  cannot  be  assigned  or  determined.  "  Such 
intermediate  magnitudes,  in  Jurin's  opinion,  cannot 
be  "  represented  to  the  mind,"  but  their  ratio  can 
be  represented  to  the  mind,  by  contemplating  the 
ratio,  *'not  in  the  vanishing  quantities  themselves, 
but  in  other  quantities  permanent  and  stable, 
which  are  always  proportional  to  them  "  (p.  'jG). 
As  to  Newton's  Lemma  i  in  Section  i,  Book  I  of 
the  Principia^  if  the  great  author  meant  to  conclude^ 
that  the  quantities  '*approach  nearer  than  any 
given  difference,"  then  he  first  supposed  what  he 
would  prove,  and  proved  only  what  he  had  before 
supposed.  Of  this  he  could  not  be  guilty.  Besides, 
Newton's  words,^  "  fiunt  eequales,"  do  absolutely 
subvert  such  an  interpretation.  Jurin  says  that 
he  does  not  claim  that  coincidence  is  necessary  for 
rigorous  proof;  he  admits  that  in  Robins's  treat- 
ment of  prime   and  ultimate  ratios,    coincidence  is 

1  Newton's  words  are  "  fiunt  ultimo  aequales."     See  our  §  4. 


no  LIMITS   AND   FLUXIONS 

not  necessary  ;  only,  Robins's  method  is  not  that  of 
Newton.  To  establish  this  last  point,  Philalethes 
quotes  Newton's  lemma  in  Latin,  then  prints  Robins's 
and  his  own  translation  of  it.  In  case  of  variation, 
the  upper  line  is  Robins's  translation,  the  lower  is 
Jurin's  : — 

Quantities ^  and  \    j     \   ratio's  of  quantities,    that 

,     .  ^    -.   .-  .     .7   {approach  eacÌLOthevA 

ciurins!;  any  finite  time  constantlyX  ,  ,.         J- 

^      -"-^  -^  \     tend  to  equahty ,     J 

and  before    the   end  of    that    time   approach    nearer 

i  than  any  given  difference,  are  ultimately  equal.  \ 

\  to  one  another  than  to  have  any  given  difference^  do  \ 

at  last  become  equal.  ) 

It  is  not  clear  to  Jurin  what  Robins  means  by 
**are  ultimately  equal,"  nor  can  Jurin  conceive 
"how  quantities,  which  do  never  become  actually 
equal,  .  .  .  can  come  within  the  description  of  a 
Lemma,  which  Lemma  expressly  affirms,  that  they 
become  equal.  "  Fiunt  ultimo  cequales  means  *  *  become 
at  last  equal."  Jurin  quotes  different  places  in  the 
Principia  which  support  his  point.  He  denies  that 
Newton  proceeds,  in  the  case  of  inscribed  and 
circumscribed  rectangles,  by  continuai  divisions  of 
the  base  of  the  figure,  and  gives  references  in 
support  of  his  contention.  Of  interest  are  the 
foUowing  admissions  made  by  Jurin  (p.  87):  ''This 
equality  therefore  we  are  obliged  to  acknowledge, 
although  we  should  not  be  able,  by  stretch  of 
imagination,  to  pursue  these  figures,  and,  as  it 
were,   to   keep  them  in  sight  ali  the  way,    till  the 


JURIN  V.   ROBIN S  AND  PEMBERTON    ni 

very  point  of  time  that  they  arrive  at  this  equality. 
For  a  demonstrated  truth  must  be  owned,  though 
we  do  not  perfectly  see  every  step  by  which  the 
thing  is  brought  about." 

*'We  have  therefore  no  occasion  for  the  delinea- 
tion  of  a  line  less  than  any  line  that  can  be  assigned. 
We  acknowledge  such  delineation  to  be  utterly 
impossible  ;  as  likewise  the  conception  of  such  a 
line,  as  an  actually  existing,  fixed,  invariable, 
determinate  quantity,"  Jurin  here  begins  to  dis- 
avow  infinitesimals.  'M  am  very  free  to  own  that 
Sir  Isaac  Newton  does  not  always  consider  this 
coincidence,  or  rather  equality,  of  the  variable 
quantity,  or  ratio  and  its  ultimate,  as  necessary  in 
his  method." 

128.  The  debate  between  Jurin  and  Robins  had 
reduced  itself  by  this  time,  not  so  much  to  the 
discussion  of  the  logicai  foundations  of  fluxions,  as 
to  the  discussion  of  what  Newton's  own  views  on 
the  subject  had  been.  Robins  prepared  a  long 
paper  on  the  subject  for  the  Aprii,  1736,  issue  of 
the  Republick  of  Letters^  under  the  title  :  A  Dis- 
sertation  shewing,  that  the  Account  of  the  doctrine  of 
Fluxions,  and  of  prime  and  ultimate  ratios,  delivered 
in  a  treatise  entitledy  '  A  discourse  concerning  the 
nature  and  certainty  of  Sir  Isaac  Newton's  methods 
of  fluxions,  and  of  prime  and  ultimate  ratios,  '  is 
agreeable  to  the  real  sense  and  meaning  of  their  great 
inventor.  The  paper  covers  45  pages.  Robins 
repeats  the  fundamental  definitions  and  historical 
statements  given  in  his  earlier  papers,  and  directs 


112  LIMITS   AND   FLUXIONS 

some  attacks  against  Berkeley,  To  set  forth  the 
views  of  Newton,  quotations  are  made  from  his 
Works,  He  quotes  from  the  Introduction  to  the 
Quadratura  Cuì^ai-um  (see  our  §§  27-42).  From 
the   Quadratura  Cuì-varum  itself  he  quotes: 

*' Quantitates  indeterminatas,  ut  motu  perpetuo 
crescentes  vel  decrescentes,  id  est,  ut  fluentes  vel 
defluentes,  in  sequentibus  considero,  designoque 
literis  z,  j'y  X,  v,  et  earum  fluxiones,  seu  celeritates 
crescendi  noto  iisdem  Hteris  punctatis.  Sunt  et 
harum  fluxionum  fluxiones,  sive  mutationes  magis 
aut  minus  celeres,  quas  ipsarum  ^,  j,  ;r,  v  fluxiones 
secundas  nominare  licet,"  etc. 

Robins  quotes  also  from  the  anonymous  account 
of  John  ColHns's  Commerdum  Epistolicum,  which 
figures  so  prominently  in  the  controversy  between 
the  foUowers  of  Newton  and  of  Leibniz.  This 
account  was  published  in  the  Philosophical  Tì'ans- 
actions,  voi.  xxix,  for  the  years  17 14,  1715,  17 16, 
of  the  Royal  Society  of  London,  of  which  Robins 
was  a  member.'  Robins  goes  on  the  assumption 
that  the  anonymous  article  was  written  by  Newton 
himself,  an  assumption  denied  by  no  one  at  that 
time  or  since,  though  Jurin  in  a  reply  wants  to 
know  on  what  authority  Newton's  authorship  is 
asserted.      Robins  quotes  as  follows  (see  our  §  47): 

*'When  he  [Newton]  considers  lines  as  fluents 
described  by  points,  whose  velocities  increase  or 
decrease,  the  velocities  are  the  first  fluxions,  and 
their  increase  the  second.  " 

129.    Robins  says  that  Berkeley,  '*for  the  support 


JURIN  V.   ROBINS  AND  PEMBERTON    113 

of  his  objections  against  this  doctrine  [of  fluxions], 
found  it  necessary  to  represent  the  idea  of  fluxions 
as  inseparably  connected  with  the  doctrine  of 
prime  and  ultimate  ratios,  intermixing  this  plain 
and  simple  description  of  fluxions  with  the  terms 
used  in  that  other  doctrine,  to  which  the  idea  of 
fluxions  has  no  relation  :  and  at  the  same  time 
by  confounding  this  latter  doctrine  with  the 
method  of  Leibniz  and  the  foreigners,  has  proved 
himself  totally  unskill'd  in  both.  These  two 
methods  of  Sir  Isaac  Newton  are  so  absolutely 
distinct,  that  their  author  had  formed  his  idea  of 
fluxions  before  his  other  method  was  invented,  and 
that  method  is  no  otherwise  made  use  of  in  the 
doctrine  of  fluxions,  than  for  demonstrating  the 
proportion  between  different  fluxions.  For,  in  Sir 
Isaac  Newton's  words  [see  our  §§  29,  36],  as  the 
fluxions  of  quantities  are  nearly  proportional  to 
the  contemporaneous  increments  generated  in  very 
small  portions  of  time,  so  they  are  exactly  in  the 
first  ratio  of  the  augmenta  nascentia  of  their  fluents. 
With  regard  to  this  passage  the  writer  of  the 
Analyst  has  made  a  two-fold  mistake.  First,  he 
charges  Sir  Isaac  Newton,  as  saying  these  fluxions 
are  very  nearly  as  the  increments  of  the  flowing 
quantity  generated  in  the  least  equal  particles  of 
time.  Again,  he  always  represents  these  augmenta 
nascentia,  not  as  finite  indeterminate  quantities,  the 
nearest  limit  of  whose  continually  varying  pro- 
portions  are  here  called  their  first  ratio,  but  as 
quantities  just  starting  out  from  non-existence,  and 

8 


114  LIMITS    AND   FLUXIONS 

yet  not  arrived  at  any  magnitude,  like  the  infini- 
tesimals  of  differential  calculus.  But  this  is  con- 
trary  to  the  express  words  of  Sir  Isaac  Newton, 
who  after  he  had  shewn  how  to  assign  by  his 
method  of  prime  and  ultimate  ratios  the  proportion, 
that  difìerent  fluxions  have  to  one  another,  he  thus 
concludes.  In  finitis  autem  quantitatilms  Analysin 
sic  institueì'e  et finitarum  nascentium  vel  evafiescentiufn 
rationes  p7'imas  vel  ultimas  investigare  consonum  est 
geo7netrice  veteì-uni:  et  volui  estendere^  quod  in  methodo 
fluxionum  non  opus  sit  figuras  infinite  pai-vas  in 
geometriam  introducei-e.''     (See  our   §§   33,  41.) 

130.  Robins  proceeds  to  explain  that  the  method 
of  prime  and  ultimate  ratios  is  ''no  other.than  the 
abbreviation  and  improvement  of  the  form  of 
demonstrating  used  by  the  ancients  on  the  like 
occasions. "  It  has  nothing  to  do  with  infinitely 
small  quantities,  which  have  led  into  error  not  only 
Leibniz  in  studying  the  resistance  of  fluids  and  the 
motion  of  heavenly  bodies,  but  also  Bernoulli  like- 
wise  in  the  resistance  of  fluids  and  in  the  study  of 
isoperimetrical  curves.  Such  infinitely  small  quanti- 
ties led  Parent  to  make  wrong  deductions.  It  was 
argued  that  because  a  heavy  body  descends  through 
the  chord  of  a  circle  terminating  at  its  lowest  point 
in  the  same  time  as  along  a  vertical  diameter,  "  the 
time  of  the  fall  through  the  smallest  arches  must  be 
equal  to  the  time  of  the  fall  through  the  diameter." 
To  relieve  Newton  of  the  suspicion  of  not  being 
free  from  the  obscure  methods  of  indivisibles, 
Robins     says    he     [Robins]    defined    an    **  ultimate 


JURIN  V.   ROBIN S  AND  PEMBERTON    115 

magnitude  "  and  **  ultimate  ratio"  as  limits.  This 
exposition  Robins  had  given  in  full  in  his  Discourse. 
The  difference  of  interpretation  of  Newton's  Lemma 
I  in  the  Principia  (Book  I,  Section  i),  given  by  him- 
self  and  by  Jurin,  arises  from  Jurin's  misinterpre- 
tation  of  Newton's  word  given.  He  **supposes  it 
to  stand  for  assignable\  whereas  it  properly  signi- 
fies  only  what  is  actually  assigned."  Jurin  claims 
that  by  our  interpretation,  ''Newton  is  rendred 
obnoxious  to  the  charge  of  first  supposing  what 
he  would  prove  "  (p.  307).  Robins  says  in  reply 
that  the  statement,  quantities  which  **are  perpetu- 
ally  approaching  each  other  in  such  a  manner,  that 
any  difference  how  minute  soever  being  given,  a 
finite  time  may  be  assigned,  before  the  end  of 
which  the  difference  of  those  quantities  or  ratios 
shall  become  less  than  that  given  difference,"  is  an 
obvious  but  not  an  identical  proposition.  Robins 
argues,  "that  Sir  Isaac  Newton  had  neither 
demonstrated  the  actual  equality  of  ali  quantities 
capable  of  being  brought  under  this  lemma,  nor 
that  he  intended  so  to  do  "  (p.  309)  ;  when  quanti- 
ties "are  incapable  of  such  equality,  the  phrase  of 
ultimately  equal  must  of  necessity  be  interpreted  in 
a  somewhat  laxer  sense,"  as  in  Principia^  Book  I, 
Prop.  71,  "prò  ajqualibus  habeantur,  are  to  be 
esteemed  equal."  When  Newton  says  that  the 
number  of  inscribed  parallelograms  should  be 
augmented  in  infinituDi^  he  does  not  mean  that 
the  number  bccomes  infinitely  great,  but  that  they 
are  augmented  endlessly.      The  nature  of  the  motion 


ii6  LIMITS  AND   FLUXIONS 

assumed  by  Jurin  to  explain  how  the  limit  may  be 
reached  is  excessively  complex.  Moreover,  ''to 
assert  that  any  collection  of  these  inscribed  and 
circumscribed  parallelograms  can  ever  become 
actually  equal  to  the  curve,  is  certainly  an  impro- 
priety  of  speech,  .  .  .  the  essence  of  indivisibles 
consists  in  endeavouring  to  represent  to  the  mind 
such  inscribed  or  circumscribed  figure,  as  actually 
subsisting,  equal  to  the  curve"  (p.  312).  Our 
interpretation  "  thus  removes  this  doctrine  quite 
beyond  the  reach  of  every  objection  "  (p.  315). 
Robins  argues  that  Newton's  ultimce  rationes^ 
quibuscuni  quantitates  evanescunt  are  not  rationes 
quantitatum  ultìmaruin  ;  but  only  limits,  to  which 
the  ratios  of  these  quantities,  which  themselves 
decrease  without  limit,  continually  approach  ;  and 
to  which  these  ratios  can  come  within  any  differ- 
ence,  that  may  be  given,  but  never  pass,  nor  even 
reach  those  limits"  (p.  316).  ''Newton  has 
expressly  told  us,  that  the  quantities,  he  calls 
nascentes  and  evanescentes,  are  by  him  always  con- 
sidered  as  finite  quantities"  (p.  321). 

131.  The  7nome?ita  of  quantities  occur  in  Newton's 
De  analysi per  cBquationes  numero  terminoìmm  infinitas, 
drawn  up  in  1666.  Newton  says  "that  he  there 
called  the  moment  of  a  line  a  point  in  the  sense  of 
Cavalerius,  and  the  moment  of  an  area  a  line  in  the 
same  sense  "  (see  our  §  47),  that  "  from  the  moments 
of  time  he  gave  the  name  of  moments  to  the  momen- 
taneous  increases,  or  infinitely  small  parts  of  the 
abscissa  and  area  generated  in  moments  of  time  .  .  . 


JURIN  V.   ROBINS  AND  PEMBERTON    117 

because  we  have  no  ideas  of  infinitely  little  quantities, 
he  introduced  fluxions  into  his  method,  that  it  might 
proceed  by  finite  quantities  as  much  as  possible." 
Prime  and  ultimate  ratios  he  introduced  later. 
Newton  says  in  that  place  that  in  his  proofs  he 
uses  0  for  a  finite  moment  of  time,  though  some- 
times,  for  brevity,  he  supposes  0  infinitely  little. 
Thus  Newton  used  0  in  two  senses  ;  in  the  fluxions 
published  in  1693  in  Wallis's  algebra,  0  is  used  in 
the  sense  of  indivisibles  ;  in  1704  he  gave  it  a 
second  signi fication  in  the  Quadratura  Curvarum. 
Robins  sums  up  his  dissertation  thus:  **  Hence  it 
is  very  manifest,  that  as  Sir  Isaac  Newton  used  at 
first  indivisibles,  so  he  soon  corrected  those  faulty 
notions  by  his  doctrine  of  fluxions,  and  afterwards 
by  that  of  prime  and  ultimate  ratios.  And  ali  the 
absurdity  of  expression,  and  ali  the  inconsistency 
with  himself  charged  on  him  by  the  author  of  the 
Analyst,  arises  whoUy  from  mis-representation." 
This  paper  was  badly  arranged  and  below  the  level 
of  Robins's  earlier  contributions. 

132.  Robins's  long  paper  in  the  Republick  of 
Letters  was  followed  in  the  July  and  August 
(voi.  xxviii,  1736)  numbers  by  Considerations 
upon  some  passages  of  a  Dissertation  conce rning  the 
Doctrine  of  Fluxions^  published  by  Mr.  Robins  in  the 
Republick  of  Letters  for  Aprii  last,  by  Philalethes 
Cantabrigiensis.  The  paper  extends  over  136  pages, 
and  could  not  be  easily  accommodated  in  a  single 
number.  From  now  on  the  disputants,  particularly 
Jurin,  are  no  longer  in  a  calm  frame  of  mind.   .  The 


ii8  LIMITS    AND   FLUXIONS 

debate  is  one  over  words  and  ceases  to  be  illumi- 
nating.  Their  judgments  were  perverted  by  the 
heat  of  controversy.  Even  theological  or  politicai 
controversies  could  not  easily  surpass  the  verbosity 
and  haze  exhibited  here. 

Jurin's  first  objection  to  Robins's  last  analysìs  is 
the  statement  that  the  method  of  fluxions  has  no 
relation  to  the  method  of  first  and  last  ratios  ;  Jurin 
quotes  from  Newton  in  support  of  his  contention. 
The  charge  that  he  (Jurin)  represents  augmentia 
nascentia  not  as  finite,  but  as  just  starting  out  of 
non-existence,  'Mike  infinitesimals  of  the  differential 
calculus,"  Jurin  denies,  saying  :  Leibniz's  differ- 
entials  **are  fixed,  determinate,  invariable  "  ;  he 
himself  has  represented  the  nascent  augments  as 
"quantities  just  starting  out  from  non-existence, 
and  yet  not  arrived  at  any  magnitude,  and  not  as 
finite  quantities"  (p.  52),  and  quotes  Newton  in 
support  of  this  view.  According  to  the  article  in 
the  Pìiilosophical  Transactions,  No.  342,  attributed 
to  Newton,  moments  are  represented  "  by  the 
rectangles  under  the  fluxions  and  the  moment  0  "  ; 
"in  his  calculus  there  is  but  one  infinitely  little 
quantity  represented  by  a  symbol,  the  symbol  0  :  it 
is  also  said,  Prick'd  letters  never  signify  moments, 
unless  when  they  are  multiplied  by  the  moment  0 
either  exprest  or  understood  to  make  them  infinitely 
little,  and  then  the  rectangles  are  put  for  moments." 
Jurin  charges  that  Robins  has  now  published  four 
different  interpretations  of  Newton's  much-discussed 
lemma.      Newton's  phrase,  jfìunt  ultimo  cequales,  the 


JURIN  V.   ROBINS  AND  PEMBERTON    119 

use  of  the  words  **  perpetually  "  and  **  endlessly," 
''the  last  difference,"  are  again  discussed  at  length. 
Jurin  quotes  from  Robins  a  passage  which  appears 
to  show  that  "  Mr.  Robins  is  now  of  opinion,  that 
Sir  Isaac's  demonstration  is  appHcable  to  such 
quantities,  as  at  last  become  actually  equal,  as  well 
as  to  quantities,  which  only  approach  without  Hmit 
to  the  ratio  of  equaHty  "  (p.  6"])  ;  therefore,  the 
lemma,  ''by  Mr.  Robins's  own  confession,  may  be 
taken  in  the  sense  1  have  always  understood  it  in  " 
(p.  68).  However,  this  is  in  direct  conflict  with 
Robins's  earlier  assertions.  In  the  discussion  about 
the  inscribed  rectangles,  both  Robins  and  Jurin 
agree  that  if  the  *  '  base  of  the  curve  "  (our  abscissa) 
be  continually  subdivided  as  in  Euclid  I  io  or  V 
IO,  it  is  manifest  ''that  such  subdivision  can  never 
be  actually  finished  "  (p.  78)  ;  but  Newton  proceeds 
difìferently — he  supposes  a  line  to  be  described  by  a 
moving  point.  Jurin  thereupon  repeats  exactly  the 
argument  in  Zeno's  "  Dichotomy,"  though  he  does 
not  mention  Zeno,  to  show  that  a  point  moving 
across  the  page  in,  say,  one  hour  passes  over  i  /  2 
of  the  distance,  then  over  i  /  4  of  it,  then  over 
1/8,  i  /  16,  etc. ,  and  insists  that  "ali  the  possible 
subdivisions  of  the  line  "  will  be  "  actually  finished  " 
and  "  brought  to  a  period  at  the  end  of  the 
hour."  This  is  given  in  support  of  bis  previous 
argument  that  the  rectangles  inscribed  in  a  curve 
may  reach  the  limit.  "  If  Mr.  Robins  will  teli  me, 
that  the  imagination  cannot  pursue  these  parallelo- 
grams  to  the  very  end  of  the  hour,  I  may  ask  him, 


I20 


LIMITS    AND   FLUXIONS 


whether  the  imagination  can  any  better  pursue  the 
subdivision  of  the  line,  or  even  of  the  hour  itself,  to 
the  end  of  the  hour,  vvhìch  subdivisions  he  must 
own  to  be  brought  to  a  period  by  the  end  of  the  hour. 
But  there  is  no  need  to  strain  our  imagination^  to 
labour  in  every  case,  or  indeed  in  any  case,  after 
some  idea  of  motion  however  intricate  ;  no  subtle 
inquiry  is  at  ali  necessary,  since  we  are  obliged  to 
own  the  conclusion  to  be  true  and  certain.    ..." 

"  However,  since  Mr. 
Robins  is  pleased  to  talk  so 
much  about  straining  our 
imagination,  .  .  .  let  us  see, 
if  we  cannot  find  some  plain 
and  easy  way  of  represent- 
ing  to  the  imagination,  that 
actual  equality,  at  which  the 
inscribed  and  circumscribed 
figures  will  arrive  with  each 
other,  and  with  the  curvi- 
linea! figure,  at  the  expiration  of  the  finite  time  " 

(p.  III). 

Let  the  curvilineal  figure  ABE  equal  in  area  the 
rectangle  with  sides  EA  and  AF.  When  the  moving 
point  describing  the  base  EA  in  a  finite  time  is  at  C, 
let  the  rectangle  with  the  base  EA  and  height  Cd 
be  equal  to  the  sum  of  the  parallelograms  inscribed 
in  ABE  (not  drawn  in  the  figure)  which  stand  on 
CA  and  upon  as  many  other  adjoining  parts  of  EA 
as  can  be  taken  equal  to  CA.  Let  lidd  be  the  curve 
traced  by  the  moving  point  d. 


eE 


JURIN  V.   ROBINS  AND  PEM BERTO N    121 

Let  the  area  of  the  rectangle  with  EA  as 
base  and  CD  as  height  =  sum  of  the  circumscribed 
parallelograms  (not  drawn  in  the  figure)  standing  on 
CA  and  upon  as  many  other  parts  of  EA  as  can  be 
taken  equal  to  CA  and  adjoining  to  it  ;  also  let 
GDD  be  the  curve  traced  by  the  movable  point  D. 
Then  as  the  curvilineal,  the  inscribed,  and  the 
circumscribed  figures  are  respectively  equal  to  EA  x 
AF,  EA  X  C<^,  EA  x  CD,  these  figures  must  be  pro- 
portional  to  AF,  C<:/,  and  CD.  These  three  lines 
will  '  '  be  equal  to  one  another  at  the  end  of  the 
finite  time."  Now  since  C<:/ and  CD  approach  each 
other,  during  a  finite  time,  within  less  than  any 
given  distance  before  the  end  of  that  time,  these 
three  lines  will,  by  that  Leniina,  be  equal  to  one 
another  at  the  end  of  the  finite  time.  The  limit  is 
reached  (p.   1 14). 

133.  As  a  further  illustration,  Jurin  takes  a 
rectilinear  figure,  the  right  triangle  ABE,  where 
EA  =  AB  =  ^,  AF=J^,  EC=;r,  the  point  C  mov- 
ing  from  E  to  A  as  before.  Upon  AC  as  a  base, 
imagine  an  inscribed  rectangle  (height  CH),  and 
a  circumscribed  rectangle  (height  CK).  As  in 
the  previous  figure,  imagine  other  inscribed  and 
circumscribed  rectangles,  standing  upon  as  many 
other  parts  of  EA  as  can  be  taken  equal  to  CA, 
and  adjoining  to  it  in  order.  When  CA  is  an 
aliquot  part  of  AE,  then  a  x  Cd  is  the  sum  of  the 
inscribed  rectangles  and  a  x  CD  is  the  sum  of  the 
circumscribed  rectangles,  where  Cd—x  /  2,  and 
C'D  =  a-x  I  2.      Let  K^/=CD.      The  ordinate  Kd, 


122 


LIMITS    AND   FLUXIONS 


B     K 


K 


N 
< 

^ 

^ 

D 

c       e 

FiG.  6. 


e  E 


drawn  to  the  base  BG,  will  be  terminate  by  EF. 
When  CA  is  not  an  aliquot  part  of  AE,  if  we 
divide  the  base  into  as  many  parts  as  may  be, 
there  will  be  left  a  portion  Eé?,  which,  let  us 
cali  r.  Then  Qd=x-\-rv.a  —  r^{2à)  and  ali  these 
ordinates  will  be  bounded  by 
Y.ddY.  In  the  same  way, 
Y^d  =a—x-j-rxa-~r-^  (2«), 
and  the  ordinate  will  be 
bounded  by  EddF.  When 
x=a,  r  vanishes,  Cd=  |  a  and 
Kd=  la.  Hence  the  inscribed 
and  circumscribed  figures  do 
then  become  equal  to  each 
other,  and  to  the  triangle 
ABE  ;  again,  the  limit  is  reached. 

Jurin  takes  Robins  to  task  for  asserting  that 
**  equality  can  properly  subsist  only  between  figures 
distinct  from  each  other."  To  Robins's  query, 
*'Does  Philalethes  here  suppose  the  truth  of  Sir 
Isaac  Newton's  demonstrations  to  depend  on  this 
actual  equality  of  the  variable  quantity  and  its 
limit?"  Jurin  answers,  "  I  do  .  .  .  In  the  manner 
Mr.  Robins  defines,  and  treats  of  prime  and  ultimate 
ratios,  I  allow  his  demonstrations  to  be  just  without 
this  actual  equality.  But  Sir  Isaac  Newton  does 
not  define  and  treat  of  prime  and  ultimate  ratios, 
in  the  same  manner  with  Mr.  Robins  ;  nor  are 
Mr.  Robins's  demonstrations  at  ali  like  Sir  Isaac 
Newton's  demonstration  "  (p.  128).  The  inability 
of  our  imagination  to  pursue  the  rectangles  in  reach- 


JURIN  V.   ROBINS  AND  PEMBERTON    123 

ing  the  limit  is  no  valid  argument  against  the  con- 
tention  that  the  limit  is  reached  ;  even  in  the 
ancient  geometry  there  are  demonstrated  truths 
that  He  beyond  the  reach  of  the  imagination,  as 
for  instance,  that  three  cones  may  equal  a  cylinder, 
ali  of  the  same  base  and  height  (p.  130).  The 
meaning  of  moment^  a  truly  diffìcult  concept,  is  dis- 
cussed  again,  Jurin  holding  that  Newton  took  it  as 
*'a  mcmentaneous  increment,  .  .  .  less  than  any 
finite  quantity  whatsoever,  and  proportional  to  the 
velocity  of  the  flowing  quantity,"  while  Robins  seem- 
ingly  claimed  that  Newton  meant  them  to  be  finite 
quantities  (p.  151).  With  respect  to  Newton's 
early  use  of  the  infinitely  little,  Jurin  and  Robins 
were  in  disagreement,  and  Robins  was  in  our 
opinion  nearer  the  truth.  Robins  claimed  that 
Newton  at  first  used  infinitely  little  quantities  ;  that 
afterwards  he  improved  his  method  by  discarding 
them  ;  Jurin  claimed  that  Newton's  alleged  absurdity 
of  expression  and  inconsistency  with  himself,  as 
charged  by  Berkeley  and  others,  **arises  wholly 
from  misinterpretation,   or  misunderstanding  him  " 

(P.   179). 

134.  Jurin's  article  appeared  in  the  July  and 
August  numbers,  1736,  of  the  Republick  of  Letters. 
Robins  could  not  wait  in  patience  until  the  entire 
article  of  Jurin  had  been  printed.  In  the  August 
number  he  replies  to  the  part  of  Jurin's  article  that 
had  appeared  in  the  July  number.  The  August 
number  was  given  up  to  Jurin  and  Robins,  to  the 
entire  exclusion  of  ali  othcr  articles  and  of  the  usuai 


124  LIMITS    AND   FLUXIONS 

hook  reviews.  On  the  last  page  of  the  August 
number,  the  editor  apologises  to  the  readers  and 
assures  them  *  *  they  shall  hereafter  have  no  occa- 
sion  to  complain  upon  this  head."  In  Robins's 
reply,  ^  both  ''Robins"  and  '*  Fhilalethes  "  appear 
in  the  third  person,  as  if  the  writer  were  some  out- 
sider. Robins  says  :  '  *  Newton  does  not  intermix 
his  simple  and  plain  description  of  fluxions  with  the 
terms  used  in  the  doctrine  of  prime  and  ultimate 
ratios  ;  for  his  description  of  fluxions  is  contained 
in  the  two  first  paragraphs  of  his  Introduction  to 
the  Quadratures,  in  which  no  terms  of  the  other 
doctrine  occur  "  (p.  89).  The  Lemma  is,  of  course, 
taken  up  again,  Robins  claiming  his  interpretation 
legitimate,  "  for  two  quantities  may  constantly  tend 
to  equality  during  some  finite  space  of  time,  and 
before  the  end  of  that  time  come  nearer  together 
than  to  have  any  difference,  which  shall  be  given  ; 
and  yet  at  the  end  of  that  time  have  stili  a  real 
difference,"  while  Jurin's  interpretation  was  not 
"any  difference  that  shall  be  given,"  but  '*any 
assignable  difference,"  which  would  mean  that  the 
limit  must  be  reached.  Mr.  Robins  says  (p.  97)  : 
'*  It  is  not  difificult  to  assign  a  very  probable  reason, 
which  led  Sir  Isaac  Newton  to  the  use  of  this 
expression  [fiunt  ultimo  aequales],  for  before  him  it 
had  not  been  unusual  for  geometers  to  speak  of  the 
last  sums  of  infinite  progressions,  which  is  an  ex- 

^  "  Remarks  on  the  Considerations  relating  to  Fluxiojis,  eie,  that 
were  published  by  Philalethes  Cantabrigiensis  in  the  Republick  of 
Letters  for  the  last  month,"  Republick  of  Letters,  August,  1736, 
pp.  87-110. 


fURIN  V.  ROBINS  AND  PEMBERTON    125 

pression  something  similar  to  this.  Surely  here  no 
one  vvill  pretend,  that  an  infinite  number  of  terms 
can  in  strict  propriety  of  speech,  and  without  a 
figure,  be  said  to  be  capable  of  being  actually 
summed  up  and  added  together.  "  Robins  makes 
the  only  direct  reference  that  was  niade  in  this 
debate  to  Zeno's  paradoxes.  He  mentions  Achilles 
and  the  Tortoise,  but  in  a  manner  devoid  of  interest. 
Referring  to  the  line  which  Jurin  supposes  traced  in 
one  hour,  Robins  says  :  "  Perhaps  it  may  be  easiest 
understood  by  comparing  the  present  point  with 
the  old  argument  against  motion  from  Achilles  and 
the  Tortoise.  It  is  impossible  to  pursue  in  the 
imagination  their  motion  by  the  means  proposed 
in  that  argument  to  the  point  of  their  meeting, 
because  the  motion  of  each  is  described  by  the 
terms  of  an  infinite  progression."  Robins  does 
not  seem  fully  to  realise  that  Achilles  and  the 
Tortoise  present  a  case  in  which  a  variable  reaches 
its  limit. 

135.  The  editor  of  the  Republick  of  Letters 
permitted  the  two  disputants  to  continue  their 
wranglings  in  an  Appendix  to  the  September  issue.^ 

Philalethes's  attempt  to  represent  to  the  imagina- 
tion the  actual  equality  at  which  the  inscribed  and 
circumscribed  figures  will  arrive  with  each  other,  and 
with  the  curvilinear  figure,  is  criticised  by  Robins 

^  Ali  Appendix  to  the  Present  State  of  the  Republick  of  Letters  for 
the  Moìith  of  September y  1736.  Being  Remarks  on  the  Remainder  of 
the  Considerations  relating  to  Fluxions,  etc.^  that  was  piiblished  by 
Philalcthes  Cantabrigiensis  in  the  Republick  of  Letters  for  the  last 
Month.  To  which  is  added  by  Dr.  Pemberton  a  Postscript  occasioned  by 
a  passage  in  the  said  Considerations.     London,  1736. 


126  LIMITS   AND   FLUXIONS 

on  the  ground  that  the  continued  curve  *'is  not  to 
be  described,  but  by  an  endless  number  of  para- 
bola's  "  (of  which  the  curve  is  the  envelope)  ;  thus, 
Philalethes  gave  *'as  an  equation  expressing  the 
nature  of  a  single  curve,  one  which  in  reality 
includes  an  infinite  series."  "  Philalethes  supposed 
a  last  form  of  the  inscribed  figures,  that  was  equal 
to  the  curve."  Robins  observed  "that  equality 
implies  the  things,  which  bave  that  property,  to  be 
distinct  from  each  other.  For  to  say  a  thing  is 
equal  to  itself  is  certainly  no  proper  expression." 
But  ''there  is  no  such  last  form  distinct  from  the 
curve,"  as  Philalethes  admits  ;  hence  Philalethes 
**gives  up  the  point.  " 

136.  In  the  Principia^  Newton  does  not  deliver 
the  doctrine  of  fluxions,  but  the  doctrine  of  prime 
and  ultimate  ratios.  ''The  understanding  of  this 
book  is  attended  with  difificulty.  "  The  expression 
ultima  sumina  is  defective  :  **Can  any  sum  of  a  set 
of  quantities,  whose  number  is  supposed  infinite,  in 
strict  propriety  of  speech  be  called  their  last  sum  ?  " 
Later,  Robins  says  :  *'  Let  Philalethes  reconcile 
the  actual  arrivai  of  these  quantities  to  the  ratio 
supposed,  and  at  the  same  instant  vanishing  away. 
Is  not  this  saying,  that  the  two  quantities  become 
nothing,  and  bear  proportion  at  the  same  instant  of 
time  ?  "  (p.  (14)).  Philalethes  "  has  thought  himself 
unjustly  accused  by  Mr.  Robins  of  supposing  a 
nascent  increment  to  be  some  intermediate  state  of 
that  increment  between  its  finite  magnitude,  and 
its  being  absolutely  nothing.      To  bave  proved  this 


JURIN  V.   ROBINS  AND  PEMBERTON    127 

assertion  groundless  he  ought  to  have  shown,  that 
this  definition  does  not  attempt  at  describing  sudi 
an  intermediate  state"  (p.  (15)).  Robins  asserts  : 
**Whoever  has  read  Sir  Isaac  Newton's  Lectiones 
OpticcB^  and  will  deny,  that  he  has  at  any  time 
made  use  of  indivisibles,  must  be  very  much  a 
stranger  to  that  doctrine,  and  to  the  style  of  those 
writers  who  foUow  it  "  (p.  (19)).  '' What  reflexion 
is  it  upon  Sir  Isaac  Newton  to  suppose,  that  he 
made  use  of  the  methods  he  had  learned  from  others 
before  he  had  invented  better  of  his  own  :  or  that 
in  an  analysis  of  a  problem  for  dispatch  he  stili 
continued  to  make  use  of  such  methods,  when  he 
conceived  it  would  create  no  error  in  the  conclusion  ? 
Has  not  Sir  Isaac  Newton  said  this  of  himself,  and 
has  Mr.  Robins  said  anything  more?"  (p.  (15)). 
"Does  Philalethes  here  mean,  that  a  quantity  can 
become  less  than  any  finite  quantity  whatsoever, 
before  it  vanishes  into  nothing?  If  not,  then  the 
point  is  given  up  to  Mr.  Robins,  who  only  contends, 
that  vanishing  quantities  can  never  by  their  diminu- 
tion  be  brought  at  last  into  any  state  or  condition, 
wherein  to  bear  the  proportion  called  their  ultimate  : 
if  otherwise,  since  Philalethes  supposes  .  .  .  that 
it  is  nonsense,  that  it  implies  a  contradiction  to 
imagine  a  quantity  actually  existing  fixed,  deter- 
minate, invariable,  indivisible,  less  than  any  finite 
quantity  whatsoever  ;  because  this  imports  as  much 
as  the  conception  of  a  quantity  less  than  any 
quantity,  that  can  be  conceived  :  how  can  a  quantity 
supposed  to  be  less  than  any  finite  quantity  whatso- 


128  LIMITS    AND   FLUXIONS 

ever  be  rendered  more  the  object  of  the  conception 
by  being  understood  to  be  brought  into  this  con- 
dition  by  a  Constant  diminution  from  a  variable 
divisible  quantity?"  (p.  (20)).  ''Sir  Isaac  Newton 
has  introduced  into  use  the  term  moment  throughout 
the  whole  second  book  of  the  Principia^  and  for  no 
other  purpose  than  for  the  sake  of  brevity  ;  for  his 
doctrine  of  prime  and  ultimate  ratios  had  been  before 
fully  explained,  and  every  proposition  of  the  second 
book  might  bave  been  treated  on  without  the  use 
of  this  term,  though  perhaps  with  a  somewhat 
greater  compass  of  words  "  (p.  (23)).  '*  Mr.  Robins 
has  endeavored  to  defend  Sir  Isaac  Newton  both 
against  the  accusation  of  the  author  of  the  Analyst^ 
and  the  misrepresentation  of  Philalethes.  He  has 
shown,  that  Sir  Isaac  Newton's  doctrine  of  prime 
and  ultimate  ratios  has  no  connexion  with  indi- 
visibles,  and  that,  if  he  ever  allowed  himself  in  the 
use  of  indivisibles,  he  knew  that  he  did  so,  and  did 
not  confound  both  the  methods  together,  as  the 
author  of  the  Analyst  accuses  him,  and  Philalethes 
without  knowing  it  has  owned  "  (p.  (27)).  '  '  Had 
Philalethes  been  versed  in  the  ancients,  and  in  the 
later  writers  who  bave  imitated  them,  he  could 
have  been  at  no  loss  about  the  true  sense  of  data 
quavis  differentia  used  by  Sir  Isaac  Newton  in  his 
first  Lemma.  For  this  expression  is  borrowed  from 
the  writers,  that  made  use  of  exhaustions  "  (p.  (29)). 
"  What  separates  the  doctrine  of  prime  and  ultimate 
ratios  from  indivisibles  is  the  declaration  made  in 
the  Scholium  to  the  first  Section  of  the  PrÌ7icipia, 


JURIN  V.   ROBINS  AND  PEMBERTON    129 

that  Sir  Isaac  Newton  understood  by  the  ultimate 
sums  and  ratios  of  magnitudes  no  more  than  the 
limits  of  varying  magnitudes  and  ratio's  ;  and  he 
puts  the  defence  of  his  method  upon  this,  that  the 
determining  any  of  these  limits  is  the  subject  of 
a  problem  truly  geometrical.  To  insist,  that  the 
variable  magnitudes  and  ratio's  do  actually  attain, 
and  exist  under  these  limits,  is  the  very  essence  of 
indivisibles  "  (p.  (34)). 

Robins's  reply  in  the  August  and  September, 
1736,  numbers  of  the  Republick  of  Letters  is  con- 
densed  in  form,  yet  covers  61  pages,  It  is  im- 
possible  for  us  to  convey  an  adequate  idea  of 
the  amount  of  detail  entering  in  the  discussion. 
Altogether  Robins  shows  greater  willingness  to 
admit  that  Newton's  views  were  different  at  differ- 
ent  periods  in  his  career,  and  that  even  Newton 
may  be  guilty  of  modes  of  expression  that  are 
not  free  from  obscurity.  Moreover,  Robins  speaks 
in  general  with  greater  sincerity  than  his  opponent. 
But  Jurin  proves  himself  the  superior  of  Robins 
in  adhering  to  a  broader  and  more  comprehensive 
conception  of  variables  and  limits. 

Pemberton  enters  the  Debate 
137.  At  this  stage  a  new  party  enters  the  debate 
— Henry  Pemberton,  who  had  studied  medicine  and 
mathematics  at  Leyden  and  Paris,  had  been  a  friend 
of  Newton,  and  had  edited  the  third  edition  of  the 
Principia.  In  an  article  follo wing  the  one  of  Robins 
in  the  ''  Appendix  "  (August  and  September   1736), 

9 


I30  LIMITS    AND  FLUXIONS 

Pemberton  says  :  ''I  .  .  .  am  fuUy  satisfied,  that 
Mr.  Robins  has  expressed  Sir  Isaac  Newton's  real 
meaning. "  Pemberton  quotes  from  Newton's  Intro- 
duction  to  the  Quadrature  of  Curves  about  prime  and 
ultimate  ratios  (see.our  §§  33,  42),  and  then  remarks; 
**  Here  Sir  Isaac  Newton  expressly  calls  the  quan- 
titates  nascentes  and  evanescentes,  whose  prime  and 
ultimate  ratios  he  investigates,  by  the  appellation  of 
finite.  Now  I  desire  Philalethes  to  reconcile  this 
passage  with  his  notion  of  a  *  nascent  quantity 
being  a  quantity  not  yet  arrived  at  any  assignable 
magnitude  how  small  soever.  '  And  I  must  farther 
ask  Philalethes,  whether  he  has  not  here  attempted 
to  define  a  non-entity.  " 

138.  Robins's  last  article  and  Pemberton's  rash 
challenge  led  to  another  flow  of  words,  covering 
77  pages  in  the  '  '  Appendix  "  to  the  Republick  of 
Letters  for  November,  1736,  in  an  article  by  Jurin, 
entitled  Obseiuations  upon  some  Remarks  relating  to 
the  Method  of  Fluxions,  published  in  the  Republick  of 
Letters  for  August  last,  and  in  the  Appendix  to  that 
for  September. 

Jurin  insists  that  *'the  method  of  fluxions,  as  it 
is  drawn  up  by  Sir  Isaac  Newton,  could  not  possibly 
h^  formed  befoi^e  the  method  of  first  and  last  ratio's 
was  invented''  (p.  (6)). 

Robins  "  takes  no  notice  of  the  letter  0  being 
used  in  the  hook  of  Quadratures^  in  the  very  same 
sense  as  in  the  Analysis''  (p.  (8)).  "That  symbol 
never  denotes  any  quantity,  but  what,  by  a  con- 
tinuai  decrease,    becomes   infinitely   little,    i.e.    less 


JURIN  V.   ROBJNS   AND  PEMBERTON    131 

than    any    quantity,     and    at    last    vanìshes     into 
nothing"  (p.  (8)). 

"He  is  grossly  mistaken  in  thinking,  that 
quantities,  which,  before  the  end  of  a  finite  time, 
come  nearer  together  than  to  have  any  assignable 
difference,  will  therefore  become  equal  before  the 
end  of  that  time  "  (p.  (  1 2)).  '  '  I  have  clearly  proved 
in  November  and  January  last,  that  Sir  Isaac 
Newton  designed  no  quantities  or  ratio's  to  be  com- 
prehended  within  the  sense  of  this  lemma,  which 
do  not  become  actually  equal  "  (p.  (13)). 

"  Has  then  Mr.  Robins,  .  .  .  offered  to  shew, 
that  any  quantities  or  ratio's  incapable  of  an  actual 
equality  are  compared  in  this  lemma?  I  think  not" 
(p.  (22)).  In  January,  ''  I  use  the  following  words, 
'  This  determinate  proportion  of  the  finite  quantities 
a  and  e,  is  what  I  understand  by  the  proportion  of 
the  evanescent  augments.'  This,  l  say,  ought  to 
have  been  attended  to,  before  this  charge  against 
me  was  renewed  "  (p.  (24)).  As  regards  the  ratio 
between  the  inscribed  and  circumscribed  figures, 
**have  not  I  truly  expressed  it  ?  If  my  expression 
be  too  complex,  let  these  great  Geometers  shew  me 
a  simpler,  if  they  can,  and  I  will  make  use  of  that  " 
(p.  (34)).  Robins's  argument  about  the  last  form  of 
parallelograms  differing  from  the  limiting  curve  is 
defective  in  the  minor  of  the  syllogism  :  "  Things 
which  are  equal  are  distinct  from  each  other. "  *'  Is 
it,"  says  Jurin,  "  the  part  of  a  candid  and  ingenious 
adversary,  to  insist  always  upon  the  word  equal,  when 
a  more  proper  expression,  as  that  of  co  incidi  ng,  has 


132  LTMITS    AND   FLUXIONS 

been  used  by  bis  antagonist  ?  "  If  bis  argument 
is  sound,  "  it  will  bold  against  my  expression, 
tbat  tbe  figures  inscribed  and  circumscribed  do  at 
last  coincide  witb  tbe  curviHnear  figure."  Jurin 
claims  '*tbat  if  Mr.  Robins's  interpretation  of  tbe 
first  Lemma  be  admitted,  Sir  Isaac's  demonstra- 
tions,  as  tbey  now  stand,  will  not  be  accurate,  nor 
geometrically  rigorous,"  for,  "  as  tbey  now  stand, 
tbe  examples  be  bas  given  in  tbe  several  Lemmata 
of  tbe  first  Section,  are  of  sucb  quantities  and 
ratio's  only,  as  do  actually  arrive  at  tbeir  respective 
limits"  (pp.  (42)  and  (43)).  '' Mr.  Robins  and  I 
bave  been  disputing  some  time,  wbetber  Sir  Isaac 
Newton  used  indivisibles.  Tbat  Gentleman  main- 
tains  tbat  be  used  tbem  ;  and  grounds  bis  ebarge 
upon  tbe  term  infinitely  little^  wbicb  is  sometimes  to 
be  met  witb  in  Sir  Isaac  Newton's  writings  :  but  be 
does  not  explain  tbe  meaning  of  tbat  term,  wben 
used  eitber  by  Sir  Isaac,  or  by  tbe  writers  of  indi- 
visibles. I,  on  tbe  contrary,  distinctly  explain 
wbat  I  apprebend  to  be  meant  by  it,  botb  wben 
used  by  Sir  Isaac  Newton,  and  wben  used  by  tbe 
writers  of  indivisibles.  ...  I  supposed  tbe  writers 
upon  indivisibles,  by  an  infinitely  little  quantity, 
to  mean  a  quantity  actually  existing,  fixed,  deter- 
minate, invariable,  indivisible,  less  tban  any  finite 
quantity  wbatsoever  "  (p.  {JZ)).  Robins  quotes 
Pascal  and  Barrow  as  using  tbe  term  indefinite  in 
place  of  infinite^  but  tbe  writers  I  quoted  use  infinite 
and  infinitely  little.  Tbere  is  difference  of  usage 
among  followers  of  Cavalieri.      ''  It  is  not  denied, 


JURIN  V.   ROBINS  AND  PEMBERTON    133 

but  that  Sir  Isaac  Newton,  by  the  term  infinitely 
little^  meant  a  quantity  variable,  divisible,  that,  by 
a  Constant  diminution,  is  conceived  to  become  less 
than  any  finite  quantity  whatsoever,  and  at  last  to 
vanish  into  nothing.  By  which  meaning  ali  that  is 
faulty  in  the  method  of  indivisibles,  is  entirely 
avoided  ;  and  that  being  allowed,  the  rest  is  only 
a  dispute  about  a  word  "  (p.  (74)). 

Jurin  declares  in  a  ''Postscript"  that  "  to  carry 
on  two  controversies  at  once  is  more  than  I  have 
leisure  for  "  ;  later  "I  intend  to  accept  of  Dr. 
Pemberton's  invitation  "  ;  meanwhile  Jurin  inserts 
an  attestation  of  "  his  learned  friend  Phileleutherus 
Oxoniensis''  to  the  effect  that  this  friend  is  "  fuUy 
satisfied,  Mr.  Philalethes  has  expressed  Sir  Isaac 
Newton's  real  meaning."  The  language  of  this 
attestation  foUows  exactly  the  language  of  Pember- 
ton,  except  that  Philalethes,  and  not  Robins,  is 
now  declared  the  correct  interpreter  of  Newton. 

139.  In  the  December  issue,  1736,  of  the  Republick 
of  Lctters^  Robins  says  in  an  "  Advertisement  "  that 
"  since  Philalethes  has  given  loose  to  passion,"  he 
"  cannot  think  anything  farther  necessary  for  the 
satisfaction  of  impartial  readers  "  (p.  492),  and  now 
takes  *Meave  of  Philalethes,"  but  cannot  resist  a 
few  parting  shots.  Nor  could  Philalethes  resist 
making  reply  to  this  "  Advertisement  "  in  an 
"  Appendix  "  to  the  December  number,  1736,  of 
the  Republick  of  Letters,  in  which  he  expresses  regret 
"that  so  long  a  correspondence  should  end  in  dis- 
content  or  ili  humour."     Jurin  justifies  the  practice 


134  LIMITS    AND  FLUXIONS 

he  exercised  in  this  controversy  of  offering  poetry 
(usually  in  Latin)  for  the  sake  of  readers  who  are 
under  necessity /'of  exercising  their  faith,  rather 
than  their  reason  in  this  dispute,"  for  **  A  verse 
may  catch  him,  who  a  sermon  flies,  "  and  for  the 
sake  of  enlivening  the  subject  for  others,  ''who 
are  judges  of  the  dispute." 

140.  In  this  December  "  Appendix  "  Jurin  then 
contributes  A  Reply  to  Dr.  Pemberton's  Postscript^ 
which  takes  up  31  pages.  Referring  to  Newton's 
Lemma  i,  Jurin  says  that  in  his  former  expres- 
sion,  the  quantities  "come  nearer  to  equaUty 
than  to  bave  any  assignable  difference  between 
them,"  it  never  was  his  intention  to  assert  "that 
during  the  time  of  the  approach,  the  difference 
between  the  quantities  is  not  always  assignable  "  ; 
he  meant  "that,  though  they  shall  always  bave  a 
difference  during  the  finite  time,  yet,  before  the 
end  of  that  time,  their  difference  shall  become 
less  than  any  quantity  that  can  be  assigned.  And 
if  my  words  are  taken  in  this  sense,  the  Dr. 's 
objection  immediately  falls  to  the  ground  "  (p.  (24)). 
Mr.  Jurin  then  gives  a  "  demonstration  "  of  the 
foUowing  proposition  :  "  If  two  lines  (i)  tend  con- 
stantly  to  equality  with  each  other,  (2)  during  any 
finite  time,  as,  for  instance,  an  hour;  (3)  and  thereby, 
their  difference  become  less  than  any  quantity  that 
can  be  assigned,  (4)  before  the  end  of  the  hour  ; 
then,  at  the  end  of  that  finite  time,  or  at  the  end 
of  the  hour,  the  lines  will  be  equal."  As  to  Dr. 
Pemberton's  charge  that  Jurin  misinterprets  Newton's 


JURIN  V,   ROBINS  AND  PEMBERTON    135 

nascent  and  evanescent  increments,  Jurin  says  that 
he  discusseci  this  question  with  Robins.  Newton's 
words  in  the  Quadratura  Cujvarum,  viz.  finitarum 
nascentìuin  vel  evanescentìum^  may  mean  ''(i)  finite 
nascent  or  evanescent  quantities,  or  (2)  finite  quan- 
tities  when  they  begin  to  be,  or  when  they  vanish. 
But  the  former  sense  contradicts  the  second  Lemma 
of  the  second  Rook  of  the  Principia,  where  Sir  Isaac 
Newton  says,  cave  intellexeris  particulas  finitas  .  .  . 
and  indeed  it  is  contrary  to  the  whole  tenor  of  his 
doctrine."  The  second  interpretation  is  *'perfectly 
conformable  to  ali  the  rest  of  Sir  Isaac  Newton's 
Works"  (p.  (32)).  Jurin  repeats  that  a  nascent  in- 
crement  is  '*an  increment  not  yet  arrived  at  any 
assignable  magnitude,  how  small  soever.  "  To  Dr. 
Pemberton's  query,  whether  Jurin  "  has  not  here 
attempted  to  define  a  non-entity,"  Jurin  replies  that 
it  *'ought  not  to  be  called  simply  a  non-entity,  nor 
simply  an  entity.  It  is  a  non-entity  passing  into 
entity,  or  entity  arising  from  non-entity,  a  begin- 
ning    entity,    something    arising    out    of    nothing  " 

(P-  (37)). 

141.  The  discussion  is  carried  on  from  this  time 
in  a  journal  called  The  Works  of  the  Learned,  into 
which  the  Republick  of  Letters  and  another  journal 
had  merged.  In  the  February,  1737,  issue  Dr. 
Pemberton  appears  with  Some  Obsei-vations  on  the 
Appendix  to  the  Present  State  of  the  Republick  of 
Letters  for  December,  1736,  which  enjoys  the  merit 
of  brevity,  being  limited  to  only  two  pages.  Pem- 
berton  declares   that   in    Newton's    passage   in   the 


136  LIMITS    AND   FLUXIONS 

Quadratura  Curvarum^  "  Philalethes  cannot  remove 
my  objection  by  straining  one  or  two  of  the  words 
to  fit  his  sense  "  ;  Newton  meant  there  that  vanish- 
ing  quantities  should  not  he  ''otherwise  than  finite 
quantities"  (p.  157).  Moreover,  "  what  kind  of 
nothings  they  must  he,  which  with  any  propriety  can 
be  said  to  pass  into  somethings,  and  for  this  reason 
can  be  capable  of  bearing  proportions,  before  they  are 
become  anything,  certainly  requires  explanation." 

A  reply  by  Jurin  in  The  Works  of  the  Learned 
for  March,  1637,  is  kept  within  the  very  moderate 
compass  of  io  pages.  The  title  of  his  contribution 
is  The  Contents  of  Dr.  Pemberton's  Observations  pub- 
lished  the  last  month.  Nothing  bere  is  of  interest 
in  the  interpretation  of  Newton. 

Dr.  Pemberton's  reply  in  the  Aprii  issue  refers  to 
Jurin's  phrase,  *'they  come  nearer  to  equality  than 
to  bave  any  assignable  difference  between  them  "  : 
*'  My  objection  to  the  interpretation  of  Philalethes 
[in  the  Minute  Mathematician,  p.  88]  is,  that  these 
words,  which  compose  the  third  article  of  that  inter- 
pretation, in  conjunction  with  the  fourth  article  can 
have  no  other  signification,  than  that  the  quantities 
come  nearer  to  equality  than  to  have  any  difference 
between  them  before  that  point  of  time,  wherein 
they  are  supposed  by  the  second  article  to  become 
equal  ;  ali  which  amounts  to  this  inconsistency, 
that  there  is  a  time,  when  the  quantities  have  no 
difference,  and  yet  are  not  equal  "  (p.  306).  Dr. 
Pemberton  again  gives  his  endorsement  of  Robins's 
interpretation  of  Newton. 


JURIN  V.   ROBINS  AND  PEMBERTON    137 

Jurin  appears  with  a  12-page  article  in  the  May, 
1737,  number  of  The  Works  of  the  Learned,  say- 
ing  :  "He  stili  ascribes  to  my  words  a  meaning, 
which  I  have  again  and  again  utterly  disavowed  ; 
not  only  so,  but  he  changes  the  words  themselves, 
putting  any  difference  instead  of  any  assignable  differ- 
ence''  {^.  388).  As  to  the  Introduction  to  Newton's 
Quadratura  Curvarum^  "  in  that  very  Introduction 
Sir  Isaac  Newton  has  made  use  of  infinitely  little 
quantities,  in  the  sense  I  understand  them,  that  is, 
quantities  which  being  at  first  finite,  do  by  a  graduai 
diminution  at  last  vanish  into  nothing  and  conse- 
quently  must,  during  their  diminution,  become  less 
than  any  quantity  that  can  be  assigned  "  (p.  389). 
As  to  evanescent  quantities  being  entities  or  non- 
entities,  "If  this  page  were  divided  from  top  to 
bottom  into  two  equal  parts,  one  black,  and  the 
other  white,^  and  Dr.  Pemberton  were  to  ask  me, 
whether  the  middle  line,  which  divides  the  two 
parts,  were  black  or  white,  I  apprehend  it  would 
be  a  direct  answer  to  say,  it  is  neither  ;  it  cannot 
properly  be  called  either  a  black  line,  or  a  white 
line  ;  it  is  the  end  of  the  white  and  beginning  of 
the  black,  or  the  end  of  the  black  and  beginning  of 
the  white"  (p.  389).  '*I  was  apprised  that  Mr. 
Robins  had  ali  along  expressed  the  sentiments  of 
Dr.  Pemberton  "  (p.  393).  Dr.  Pemberton  stili 
refuses     to    give    his     interpretation    of    Newton's 

^  As  far  as  I  know,  Jurin  is  the  first  to  use  colour  devices  to  illustrate 
subtle  points  in  evanescent  quantity  or  in  number.  Jules  Tannery,  in 
his  Le^ons  cf  Algebre  et  cTAnalyse,  Paris,  1906,  p.  14,  uscs colour  imagery 
to  illustrate  the  discussion  of  irralional  numbers. 


138  LIMITS    AND   FLUXIONS 

Lemma,  •'  Every  body  will  be  satisfied  that  the 
true  reason  of  bis  backwardness,  is  the  fear  he  is 
under,  that  I  shall  make  good  my  promise,  in  shew- 
ing,  that  bis  explanation  is  either  a  false  one,  or, 
in  case  it  be  true,  is  to  ali  intents  and  purposes  the 
very  same  with  mine  "  (p.  396). 

In  June,  1737,  Dr.  Pemberton  replies  again,  by  re- 
peating  bis  previous  assertion  against  Philalethes's 
explanation  of  Newton's  Le^nma^  given  in  the 
Minute  Mathematician,  but  does  not  permit  him- 
self  to  be  drawn  into  giving  an  explanation  of  his 
own  of  Newton's  Lemma. 

In  Jurin's  article  in  the  July  issue,  1737,  we  read  : 
**I  did  indeed  take  notice  of  the  prudence  Dr. 
Pemberton  used,  in  passing  by  my  second  inter- 
pretation,  which  was  so  clear  and  plain,  and  was  so 
fully  illustrated  by  examples,  that  there  was  no 
possibility  of  perverting  the  sense  of  it  "  (p.  70). 
*'But  since  this  dispute,  which  began  upon  matters 
of  science,  .  .  .  unless  Dr.  Pemberton  shall  see  fit 
to  revive  it  by  giving  his  so  long  demanded  explica- 
tion,  I  shall  not  judge  it  worth  while  to  take  notice 
of  what  he  may  hereafter  write.  " 

Dr.  Pemberton  followed  with  some  Observations  in 
the  August,  1737,  number,  while  in  the  September 
number  there  appears  "  the  last  reply  of  Philalethes," 
and  in  the  October  number  the  final  answer  by 
Pemberton.  Thus  ended  a  dispute  which  had  for 
some  time  ceased  to  contain  much  of  scientific  and 
historic  value. 


JURIN  V.   ROBINS  AND  PEMBERTON    139 

Debate  over  Robins's  Review  of  Treatises  written  by 
Leonhard  Euler,  Robert  Smith,  and  Jurin 

142.  Being  in  a  somevvhat  combative  mood, 
Robins  made  attacks  upon  Euler's  treatise  on  motion, 
Dr.  Robert  Smith's  optics,and  Jurin's  essay  on  vision.^ 

Robins's  criticisms  of  Euler  concern  mainly  the 
philosophy  of  the  Calculus.  Robins  quotes  Euler's 
third  proposition,  "  That  in  any  unequal  motion 
the  least  element  of  the  space  described  may  be 
conceived  to  be  passed  over  with  an  uniform  motion," 
and  then  says,  this  ''is  not  universally  true,"  as, 
for  instance,  ''when  those  spaces  are  compared 
together,  which  a  body  acceleratcd  by  any  force 
described  in  the  beginning  of  its  motion  ;  for  the 
ultimate  proportion  of  the  first  of  two  contiguous 
spaces,  thus  described  in  equal  times  to  the  second, 
is  not  that  of  equality,  but  the  ratio  of  i  to  3, 
as  is  well  known  to  every  one  acquainted  with 
the  common  theory  of  falling  bodies  "  (p.  2).  In 
another  place  (p.  4)  Robins  argues  that  the  path 
assigned  by  Euler  to  a  certain  body  "  is  false  even 
on  the  confused  principles  of  indivisibles."  Some 
passages  in  Robins  involve  the  Leibnizian  notation 
in  the  calculus,  and  look  quite  odd  in  an  eighteenth- 
century  publication  prepared  by  a  Briton  in  Great 
Britain.  Robins  concludes  that  most  of  Euler's 
errors  "  are  owing  to  so  strong  an  attachment  to 
the  principles,  he  had  imbibed  under  that  inelegant 

^  Kemarks  on  Mr.  Euler's  IVeatise  of  Motion^  Dr.  Smith's  Covipleat 
System  of  Opttcks,  a7id  Dr.  Jnrins  Essay  upon  Distinct  and  IndistÌ7ict 
Vision.     By  Benjamin  Robins,  London,  1739. 


HO  LIMITS    AND   FLUXIONS 

computist,  who  was  his  instructor,  that  he  was 
afraid  to  trust  his  own  understanding  even  in  cases, 
where  the  maxims,  he  had  learnt,  seemed  to  him 
contradictory  to  common  sense  "  (p.  30).  This 
master  was  John  Bernoulli. 

143.  Never  losing  an  opportunity  to  engagé  in 
controversy,  Jurin  wrote  a  treatise  in  reply.  ^  We 
refer  only  to  such  parts  of  this  pamphlet,  and  the 
ones  which  followed  it,  as  bear  on  fluxions  or  the 
parties  engaged  in  the  discussions  on  fluxions. 

In  the  preface  Jurin  says  :  "  I,  it  seems,  am  the 
Reputed  Author  of  the  late  dissertations  under  the 
name  of  Philalethes  Cantabrigiensis,  and  the  other 
Gentleman  [Dr.  Robert  Smith]  is  .  .  .  suspected  of 
being  my  associate.  .  .  .  If  Dr.  Smith  were  to  teli 
Mr.  Robins,  what  he  has  often  professed  to  other 
persons,  that  he  had  no  band  in  those  papers  ;  if  to 
confirm  this  he  were  to  remind  him,  that  Philalethes 
has  declared  more  than  once,  he  wrote  alone  and 
unassisted  ;  if  I — But  what  signifies  pleading,  when 
the  execution  is  over  ?  Mr.  Robins  has  already 
vented  his  Resentment  to  the  utmost.    ..." 

144.  Not  without  interest  is  the  following  refer- 
ence  to  young  Euler  in  St.  Petersburg,  whose 
scientific  achievements  bave  been  so  very  extra- 
ordinary.  Jurin  says  that  to  make  no  reply  to 
Robins's  criticisms  "  might  be  such  a  discouragement 
to  the  hopeful  young  writer,  whose  name  is  prefixed 


^  A  Reply  to  Air.  Robins^ s  Remarks  on  the  Essay  upon  Distinct  and 
Indistinct  Vision  Published  at  the  End  of  Di-,  Smith^s  Compleat  System 
of  Opticks.     By  James  Jurin,  M.  D. ,  London,  MDCCXXXIX. 


JllRIN  V.    ROBIN S  AND  PEMBERTON    141 

to  theìr  common  labours,  and  who  possibly,  when 
he  Comes  to  study  suo  Marte,  and  to  see  with  his 
own  eyes,  or  to  meet  with  abler  instructors,  may 
make  some  figure  in  the  Learned  World,  that  pure 
humanity  induces  me  to  oblige  them  with  this  one 
Reply"(p.  54). 

145.  Of  course,  Robins  wrote  a  tract  in  reply,^ 
but  only  the  preface  of  this  tract  demands  our 
attention.  In  answer  to  the  charge  made  by  Jurin, 
that  he  (Robins)  had  conducted  the  controversy 
"with  passion  and  abuse,"  Robins  proceeds  to 
explain  their  past  relations  to  each  other. 

'*  About  six  years  since  a  pamphlet  was  publish'd 
under  the  title  of  the  Analyst  ;  in  which  the  author 
endeavors  to  shew,  that  the  doctrine  of  fluxions  in- 
vented  by  Sir  Isaac  Newton  is  founded  on  fallacious 
suppositions.  As  that  writer  had  a  false  idea  of  this 
doctrine,  ...  I  thought  the  most  effectual  method 
of  obviating  his  objections  would  be  to  explain  .  .  . 
what  Sir  Isaac  Newton  himself  had  delivered  with 
his  usuai  brevity.  .  .  .  And  with  this  view  I  pub- 
lished  a  Discourse  on  Sir  Isaac  Newton's  method 
of  fluxions,  and  of  prime  and  ultimate  ratios,  But 
in  the  mean  time  a  controversy  was  carrying  on 
between  the  author  of  the  Analyst  and  another, 
who  under  the  name  of  Philalethes  Cantabrigiensis 
had  undertaken  the  defence  of  Sir  Isaac  Newton  : 
and  as  I  at  last  perceived,  both  by  the  concessions 


^  A  Full  Confutation  of  Dr.  Jurin* s  Reply  io  the  Rcmarks  on  his 
Essay  upon  Distinct  and  Indistiiut  Vision.  By  Benjamin  Robins, 
London,  1740. 


142  LIMITS    AND   FLUXIONS 

of  Philalethes,  and  the  avowed  opinions  of  others, 
that  the  erroneous  conceptions  of  the  writer  of  the 
Analyst  on  thìs  head  were  more  prevalent  even 
amongst  those,  who  approved  of  the  method  of 
fluxions,  than  I  had  at  first  beHeved  ;  I  thought,  it 
might  be  no  unacceptable  task  more  particularly  to 
shew  those,  who  were  thus  misled,  how  irreconcile- 
able  their  opinions  were  with  the  tenets  of  Sir  Isaac 
Newton,  and  how  impossible  it  would  be  to  defend 
the  accuracy  of  his  doctrine  on  these  their  mistaken 
suppositions  ;  and  it  was  with  this  intention,  that  in 
an  account  of  my  hook  inserted  in  the  Present  State 
of  the  Republick  of  Letters,  some  of  the  errors  con- 
tained  in  the  writings  of  Philalethes  Cantabrigiensis 
were  endeavoured  to  be  obviated. 

''But  tho'  this  discourse  was  written  with  great 
caution,  and  only  mentioned  the  principles  objected 
to  without  so  much  as  naming  or  even  insinuating 
the  treatises,  from  whence  they  were  taken  ;  yet, 
as  Dr.  Jurin,  who  was  generally  reputed  the  author 
of  them,  was  one,  that  I  often  conversed  with  ;  at 
my  request,  before  this  paper  was  printed,  a  common 
friend  carried  to  him  the  manuscript,  and,  without 
pretending  to  suppose,  whether  he  was,  or  was  not 
Philalethes,  desired  him  to  read  it,  and  asked  him 
if  he  thought,  Philalethes  could  be  displeased  with 
any  thing  contained  in  it  ;  he  was  also  told  at  the 
same  time,  that  if  he  believed  any  part  of  it  could 
give  offence  to  that  gentleman,  whoever  he  were, 
it  should  be  struck  out,  or  that  1  would  even  let 
the  whole  design  fall,   if  he  desired  it. 


JURIN  V.   ROBINS  AND  PEMBERTON    143 

*'My  friend  brought  me  the  Doctor's  answer 
importing,  that  he  could  not  believe,  my  paper 
would  displease  any  one,  since,  if  the  tenets,  I 
excepted  to,  were  really  erroneous,  it  was  reason- 
able,  they  should  be  exposed  ;  and  if  otherwise,  it 
was  the  business  of  Philalethes  to  defend  them  .  .  . 
it  was  however  added,  that  I  had  in  two  places 
censured  doctrines,  whìch,  if  1  supposed  them  to 
be  the  opinions  of  Philalethes,  I  must  bave  mis- 
apprehended  him.  Now  ...  1  immediately  ex- 
punged  them,  and  published  the  remaining  part 
in  the  Republick  of  Letters  for  October  1735,  as 
an  account  of  my  hook  on  Sir  Isaac  Newton's 
method  of  fluxions,  and  of  prime  and  ultimate 
ratios. 

"  To  this  Philalethes  answered  in  the  foUowing 
month,  and  I  again  replied,  till  five  papers  were 
successively  written  in  this  controversy,  that  is, 
three  by  me,  and  two  by  him.  And  ali  this  time 
so  very  desirous  was  I  on  my  part  of  avoiding 
irritating  circumstances  .  .  .  that  I  thought  even 
the  most  intimate  friend  .  .  .  could  not  be  offended 
with  it.  .  .  .  But  alas  .  .  .  Philalethes  in  his  reply, 
part  of  which  was  published  in  the  July  foUowing, 
and  the  rest  in  the  succeeding  month,  runs  out  into 
the  most  extravagant  heats  of  passion  .  .  .  charg- 
ing  me  with  dishonestly  writing  against  the  con- 
victions  of  my  own  judgment.  .  .  .  After  so  gross 
and  unprovoked  an  abuse,  ...  I  should  surely  bave 
been  acquitted  of  any  breach  of  decency,  if  .  .  .  I 
had  sharply  exposed  his  ignorance  in  the  subjects, 


144  LIMITS    AND   FLUXIONS 

he  had  attempted.  But  I  chose,  ìf  possìble,  to 
avoid  the  ridicule  of  quarreling  on  a  matter  of 
mere  speculation  ...  I  again  requested  my  friend 
to  speak  to  Dr.  Jurin,  and  to  represent  to  him  the 
inconveniencies,  that  would  arise  from  the  persever- 
ance  of  Philalethes  in  his  rash  and  groundless 
calumny.  My  friend  accordingly  went  to  Dr.  Jurin, 
and  carried  with  him  an  answer  to  so  much  of 
Philalethes's  paper,  as  was  then  pubHshed,  and  told 
the  Doctor,  that  he  carne  to  propose  to  him  a  method, 
that  might  prevent  the  controversy  betwixt  m.e  and 
Philalethes  from  degenerating  into  a  passionate 
personal  altercation  .  .  .  that  therefore,  ifDr.  Jurin 
thought  it  expedient,  my  paper  should  be  given  to 
a  certain  gentleman,  to  whose  impartiality  and 
knowledge  of  the  subject  in  debate  no  exception 
could  be  taken  on  either  side  ;  and  that  if,  when 
that  gentleman  had  perused  it,  he  should  believe, 
I  had  in  any  instance  changed  my  opinion  from 
my  first  entering  into  this  dispute,  I  did  then 
promise  to  submit  patiently  and  without  reply  to 
any  censures  of  unfairness  and  dishonesty,  that 
Philalethes  .  .  .  should  hereafter  think  proper,  .  .  . 
[otherwise]  it  would  then  be  but  common  justice, 
that  Philalethes  should  moderate  the  remaining 
part  of  his  performance.  .  .  .  But  this  proposai  was 
rejected.  .  .  .  It  was  immediately  given  out,  that 
my  friends  had  confessed  me  to  have  been  foiled 
in  the  argument  ;  and  were  now  only  soUicitous  to 
support  me  from  the  charge  of  unfairness.  .  .  .  The 
reader  will  not  wonder,  if  I  resolved  for  the  future 


JURIN  V.   ROBIN S  AND  PEMBERTON    145 

to  treat  him  with  that  freedom,  which  his  unskilful- 
ness  authorised.    ..." 

146.  The  above  preface  constitutes  what  we  may 
cali  Robins's  apologia  prò  vita  sua.  It  seems  only 
fitting  that  Jurin  should  appear  with  a  similar  docu- 
ment.      This  he  did  in  a  long  Letter.^ 

We  make  the  following  quotations  from  Jurin 
(p.  8)  : 

**  About  five  years  ago  some  passages  in  a  paper 
of  Mr.  Robins,  were  shown  to  me  .  .  .  and  a  question 
was  put  to  me,  whether  I  should  take  it  ili,  if  those 
passages  were  printed,  it  being  intimated,  that 
Philalethes,  against  whom  they  were  designed, 
might  possibly  be  some  friend  of  mine  :  and  indeed, 
several  persons  were  then  guessed  at,  ali  of  which 
happened  to  be  my  friends.  To  this  ...  I  gave 
answer,  that  I  should  not  at  ali  take  it  ili.  But  I 
added,  that  as  I  had  read  the  controversy  between 
Philalethes  and  the  Author  of  the  Analyst,  with  some 
attention,  it  seemed  to  me  that  in  one  or  two 
passages  Mr.  Robins  imputed  opinions  to  Philalethes, 
which  .  .  .  that  gentleman  did  not  hold.  .  .  .  Also, 
I  took  notice,  that  Mr.  Robins  did  not  rightly 
explain  Sir  Isaac  Newton's  first  Lemma.  .  .  .  But 
when  1  desired  to  talk  with  Mr.  Robins  about  the 
Lemma,  before  the  papers  went  to  the  press,  as 
imagining  I  could  convince  him  that  he  was  in  the 
wrong,   answer  was   made,    that   the   question   was 


^  A  Letter  to  .  .  .  Esquire^  In  Answer  to  Mr.  Robins^s  Full  Confu- 
tation  of  the  Reply  to  his  Remarks  on  the  Essay  upon  distinct  and 
indistinct  Vision.     By  James  Jurin,  M.D.,  London,  1741. 

IO 


146  LIMITS    AND  FLUXIONS 

not  whether  I  thought  him  in  the  right  or  in  the 
wrong,  but  only  whether  I  should  take  anything 
amiss  ;  to  which  I  replied  as  before.  Upon  talking 
with  another  friend  of  Mr.  Robins  a  day  or  two 
after,  I  repeated  my  desire  to  talk  with  Mr.  Robins 
about  his  explanation  of  the  Lemma,  before  his 
papers  went  to  the  press  :  but  was  told  that  could 
not  be,  for  that  the  part  of  the  papers  where  the 
Lemma  was  spoke  of,  was  to  go  to  the  press  that 
afternoon.  ...  I  do  not  remember,  that  any  offer 
was  made  to  me  of  '  letting  the  whole  design  fall, 
if  I  desired  it.  '  Had  any  such  offer  been  made,  I 
had  at  that  time  so  much  regard  for  Mr.  Robins, 
that  I  think  I  should  at  least  have  desired  him  to 
stop  the  design,  till  he  and  I  had  examined  the 
Lemma  together,  in  order  to  prevent  his  exposing 
himself  in  the  manner  he  has  since  done.  As  to 
the  second  application  made  to  me  near  a  year 
after,  it  may  easily  be  judged,  that  I,  who  gave 
these  gentlemen  no  reason  to  think  I  had  any  in- 
fluence  over  Philalethes,  or  so  much  as  knew  who 
he  was,  could  neither  comply  with  nor  reject  their 
proposai  "  (p.  9). 

Remarks 

147.  The  debate  between  Jurin  and  Robins  is 
the  most  thorough  discussion  of  the  theory  of 
limits  carried  on  in  England  during  the  eighteenth 
century.  It  constitutes  a  refinement  of  previous 
conceptions. 

Jurin  possessed  the  more   general    conception  of 


JURIN  V.   ROBINS  AND  PEMBERTON    147 

a  limit  in  insisting  that  there  are  variables  which 
reach  their  limits.  His  interpretation  of  Newton 
on  this  point  appears  to  us  more  nearly  correct  than 
that  of  Robins  ;  Jurin's  geometrie  illustrations  of 
limit-reaching  variable,  intended  to  aid  the  imagina- 
tion,  though  as  he  admits  incapable  of  exhibiting  the 
process  "ali  the  way,"  are  nevertheless  interesting 
(see  our  §§  124,  132,  133).  The  imagination  is 
subject  to  limitations  where  the  reason  is  stili  free 
to  act. 

Robins,  and  after  him  Pemberton,  deserve  credit 
in  clearly,  openly,  and  completely  breaking  away 
from  infinitely  little  quantities,  and  from  prime  and 
ultimate  ratios.  Robins's  conception  of  a  limit  was 
narrow,  but  this  narrowness  had  certain  peda- 
gogical  advantages,  since  it  did  not  involve  a  mode 
of  advance  to  the  limit  which  altogether  tran- 
scended  the  power  of  the  imagination  to  follow  ali 
the  way  (see  our  §§  117,   118,   129,   130). 

It  is  interesting  to  observe  that  both  Jurin  and 
Robins  disavow  belief  in  the  possibility  of  a  sub- 
division  of  a  line  into  parts  so  as  to  reach  a  point — 
they  assert  "that  such  subdivision  can  never  be 
actually  finished  "  (see  our  §§  126,   132). 

Robins  discarded  the  use  of  Newton's  moinents 
in  developing  the  theory  of  fluxions  (see  our 
§§  119,   120). 

Toward  the  end  of  his  long  debate  with  Robins, 
Jurin  begins  to  disavow  infinitely  small  quantities. 
He  brings  out  the  difference  between  infinitesimals 
as  variables,    and   infinitesimals   as   constants.      He 


148  LIMITS    AND   FLUXIONS 

rejects  ali  quantity  *'fixed,  determinate,  invariable, 
indivisible,  less  than  any  finite  quantity  whatsoever," 
but  he  usually  admits  somewhat  hazily  a  quantity 
"variable,  divisible,  that,  by  a  Constant  diminution, 
is  conceived  to  become  less  than  any  finite  quantity 
whatever,  and  at  last  to  vanish  into  nothing.  "  (See 
our  §§  132,   138,   141.) 

VVhile  Berkeley's  Analyst  and  Berkeley's  replies 
to  Jurin  and  Walton  involved  purely  destructive 
criticism,  the  present  controversy  between  Jurin  and 
Robins  brought  forth  valuable  constructive  results. 
Jurin's  papers  against  Robins  are  decidedly  superior 
to  those  he  wrote  against  Berkeley,  though  here  too 
they  contained  much  that  was  not  pertinent  to  the 
subject  and  was  intended  merely  to  amuse  the 
general  reader. 


CHAPTER    V 

TEXT-BOOKS    IMMEDIATELY   FOLLOWING 
BERKELEY'S   ATTACK 

148.  The  Analyst  was  published  in  1734  ;  two  years 
later  appeared  four  books  on  fluxions.  Thus,  more 
British  text-books  on  this  subject  were  published 
in  1736  than  in  ali  the  thirty  years  preceding.  That 
the  Analyst  controversy  was  largely  the  cause  of 
this  increased  productivity  there  can  be  no  doubt. 
We  proceed  to  give  an  account  of  the  books  which 
preceded  the  publication  of  Maclaurin's  Treatise  of 
Fluxions^  1742. 

John   Colson^    1736 

149.  Newton's  Method  of  Fluxions,'^  S3.\d  to  have 
been  written  in  1671,  was  translated  and  first 
published  in  1736  by  John  Colson.  Colson  had 
been  a  student  at  Christ  Church,  Oxford,  which 
he  left  without  taking  a  degree.      He  was  appointed 

^  The  Method  of  Fhixions  and  Infinite  Stries  ;  with  its  Application 
to  the  (Jeometry  of  Curz'e- Lines.  By  the  Inventar^  Sir  Isaac  Newton,  Kt., 
Late  Presi dent  of  the  Koyal  Society.  Translated  from  the  Author^s 
Latin  Originai  not  yet  made  puhlick.  To  which  is  subjoined,  A 
Perpetuai  Commenttipon  the  Whole  Work,  .  .  .  By  John  Colson,  M.  A. 
and  F. R.S.,  Master  of  Sir  Joseph  Williamson's  free  Mathematical- 
School  at  Rochester.  London,  M.DCC.XXXVI.  This  hook  was 
reissued  in  1758. 

149 


ISO  LIMITS    AND  FLUXIONS 

master  of  a  new  mathematica!  school  founded  at 
Rochester,  and,  in  1739,  Lucasian  professor  of 
mathematics  at  Cambridge,  in  succession  to  Nicholas 
Saunderson.  Colson  was  a  man  of  great  industry 
but  only  ordinary  ability. 

In  his  preface,  Colson  refers  to  the  controversies 
on  fluxions,  and  says  that  the  defenders  as  well  as 
their  opponents  were  little  acquainted  with  Newton's 
own  exposition,  that  this  hook  now  published  for 
the  first  time  is  "the  only  genuine  and  originai 
Fountain  of  this  kind  of  knowledge.  For  what  has 
been  elsewhere  deliver'd  by  our  Author,  concerning 
this  Method,  was  only  accidental  and  occasionai" 
(p.  x).  Colson  accompanies  Newton's  hook  "with 
an  ampie  Commentary  "  and  "  particularly  with  an 
Eye  to  the  fore-mention'd  Controversy  "  (p.  x). 
Colson  in  this  preface  represents  Newton  as  hold- 
ing the  principle  "that  Quantity  is  infinitely 
divisible,  or  that  it  may  (mentally  at  least)  so  far 
continually  diminish,  as  at  last,  before  it  is  totally 
extinguish'd,  to  arrive  at  Quantities  that  may  be 
call'd  vanishing  Quantities,  or  which  are  infinitely 
little,  and  less  than  any  assignable  Quantity.  Or 
it  supposes  that  we  may  form  a  Notion,  not  indeed 
of  absolute,  but  of  relative  and  comparative  infinity  " 
(p.  xi).  Colson  opposes  "  indivisibles,"  as  also 
the  "  infinitesimal  method"  and  "infinitely  little 
Quantities  and  infinite  orders  and  gradations  of 
these,  not  relatively  but  absolutely  such  "  (p.  xii). 
He  argues  against  "  imaginary  Systems  of  infinitely 
great    and    infinitely    little    Quantities,    and    their 


TEXT-BOOKS,  1736-1741  151 

several  orders  and  properties,  which,  to  ali  sober 
Inquirers  into  mathematica!  Truths,  must  certainly 
appear  very  notional  and  visionary  "  (p.  xii),  for 
"Absolute  Infinity,  as  such,  can  hardly  be  the 
object  either  of  our  Conceptions  or  Calculations, 
but  relative  Infinity  may,  under  a  proper  regula- 
tion  "  (p.  xii).  Newton  "  observes  this  distinction 
very  strictly,  and  introduces  none  but  infinitely 
little  Quantities  that  are  relatively  so."  Colson 
answers  Berkeley's  criticism  in  the  Analyst  of 
Lemma  2,  Book  li,  in  the  Principia  in  the  follow- 
ing  manner  : — 

"Let  X  and  Y  be  two  variable  Lines.  .  .  .  Let 
there  be  three  periods  of  time,  at  which  X  becomes 
K  —  \a,  AjA-f-J^;  and  Y  becomes  V>  —  \b,  B,  and 
B  +  i^  .  .  .  Then  ...  the  Rectangle  XY  will 
become  .  .  .  KV>-\a?>-\bK^\ab,  AB,  and 
AB  +  i^B  +  i<^A  +  i^^.  Now  in  the  interval  from 
the  first  period  of  time  to  the  second  .  .  .  its  whole 
Increment  during  that  interval  is  \aY>-\-\bK  —  \ab. 
And  in  the  interval  from  the  second  period  of  time 
to  the  third,  ...  its  whole  Increment  during  that 
interval  is  \à^-\-\bh.-\-\ab.  Add  these  two  Incre- 
ments  together,  and  we  shall  bave  a^  +  b\  for  the 
compleat  Increment  of  the  Product  XY  "  (p.  xiii), 
called  the  "  Moment  of  the  Rectangle"  when  a  and 
b  are  infinitely  little. 

Another  mode  of  procedure  is  this:  ''the  Fluxions 
or  Velocities  of  increase,  are  always  proportional 
to  the  contemporary  Moments."  "  When  the  Incre- 
ments  become   Moments,  that  is,  when  a  and  b  are 


152  LIMITS   AND  FLUXIONS 

so  far  diminish'd,  as  to  become  infinitely  less  than 
A  and  B  ;  at  the  same  time  ab  will  become  infinitely 
less  than  either  a^  or  ^A  (for  ^  aV)  .  ab  -.  -.Vi  .b^  and 
^A  .  ab  \  :  A  .  a)^  and  therefore  it  will  vanish  in 
respect  of  them.  In  which  case  the  Moment  of  the 
Product  or  Rectangle  will  be  a^-{-bA,  as  before  " 
(p.  xv).  Newton,  however,  prefers  the  more  direct 
way  previously  explained. 

Proceeding  to  Newton  himself,  we  find  (on  p,  24) 
the  following  definition  :  "  The  Moments  of  flowing 
Quantities  (that  is,  their  indefinitely  small  Parts, 
by  the  accession  of  which,  in  infinitely  small  por- 
tions  of  Time,  they  are  continually  increased)  are 
as  the  Velocities  of  their  Flowing  or  Increasing. 
Wherefore  if  the  Moment  of  any  one,  as  ;r,  be  repre- 
sented  by  the  Product  of  its  Celerity  x  into  an 
indefinitely  small  Quantity  0  (that  is,  by  x  0),  the 
Moments  of  the  others  v,  j/,  z,  will  be  represented 
by  vo,  fo,  éo  ;  because  vo,  xo,  yo,  and  èo^  are  to  each 
other  as  v^  x,  y,  and  i."  On  p.  25  terms  contain- 
ing  (?  as  a  factor  "  will  be  nothing  in  respect  of  the 
rest.      Therefore  I  reject  them." 

1 50.  Colson  appended  extensive  annotations  to 
Newton's  treatise.  In  these  annotations,  p.  250, 
Colson  speaks  of  ''  smallest  particles,"  but  the  term 
'' smallest  "  does  not  occur  in  Newton's  definition. 
However,  Colson  says  that  he  does  not  mean 
**atoms"  nor  "definite  and  determinate  magni- 
tude,  as  in  the  Method  of  Indivisibles,"  but  things 
"indefinitely  small;  or  continually  decreasing,  till 

^  Here  a^  .  ab  :  :  V> .  h  means  a^  :  ab  :  -.  ^  :  b. 


TEXr-BOOKS,  1736-1741  153 

they  are  less  than  any  assignable  quantities,  and 
yet  may  then  retain  ali  possible  varieties  of  pro- 
portion  to  one  another.  Becoming  stili  more  deeply 
involved  in  the  metaphysics  of  the  subject,  Colson 
adds  "  that  these  Moments  are  not  chimerical, 
visionary,  or  merely  imaginary  things,  but  have  an 
existence  sui  generis^  at  least  Mathematically  and 
in  the  Understanding,  is  a  necessary  consequence 
from  the  infinite  Divisibility  of  Quantity,  which 
I  think  hardly  anybody  now  contests  "  (p.  251). 
This  he  qualifies,  ''perhaps  the  ingenious  Author 
of  .  .  .  The  Analyst  must  be  excepted,  who  is 
pleased  to  ask,  in  his  fifth  Ouery,  whether  it  be  not 
unnecessary,  as  well  as  absurd,  to  suppose  that 
finite  Extension  is  infinitely  divisible  "  (p.  251). 
By  ultimate  ratio  Colson  means  the  ratio  when  the 
arguments  ''become  Moments  "  (p.  255).  Fearing 
that  moments,  infinitely  little  quantities,  and  the 
like,  **may  furnish  most  matter  of  objection,"  he 
says  (p.  336)  that  the  symbol  0  at  first  represents 
a  finite  quantity,  which  then  diminishes  continually 
till  "  it  is  quite  exhausted,  and  terminates  in  mere 
nothing.  "  But  *'it  cannot  pass  from  being  an 
assignable  quantity  to  nothing  at  once  ;  that  were 
to  proceed  per  saltum,  and  not  continually  "  ;  hence 
"  it  must  be  less  than  any  assignable  quantity  what- 
soever,  that  is,  it  must  be  a  vanishing  quantity. 
Therefore  the  conception  of  a  Moment,  or  vanishing 
quantity,  must  be  admitted  as  a  rational  Notion  " 
(P-  336).  Again  :  "The  Impossibility  of  Concep- 
tion may  arise  from  the  narrowness  and  imperfection 


154  LIMITS    AND   FLUXIONS 

of  our  Faculties,  and  not  from  any  inconsistency  in 
the  nature  of  the  thing"  ;  these  quantities  "escape 
our  imagination.  "  Referring  to  imaginaries,  a  J  —  i 
in  the  solution  of  cubie  equations,  Colson  says 
(PP-  338-9)*  "These  impossible  quantities  .  .  . 
are  so  far  from  infecting  or  destroying  the  truth  of 
these  Conclusions,  that  they  are  the  necessary 
means  and  helps  of  discovering  it.  And  why  may 
we  not  conclude  the  same  of  that  other  species  of 
impossible  quantities,  if  they  must  needs  be  thought 
and  call'd  so  ?  .  .  .  Therefore  the  admitting  and 
retaining  these  Quantities  .  .  .  'tis  enlarging  the 
number  of  general  Principles  and  Methods,  which 
will  always  greatly  contribute  to  the  Advancement 
of  true  Science.  In  short,  it  will  enable  us  to  make 
a  much  greater  progress  and  proflcience,  than  we 
otherwise  can  do,  in  cultivating  and  improving  what 
I  have  elsewhere  call'd  The  Philosophy  of  Quantity,  " 
151.  A  review  1  of  this  book  contains  the  follow- 
ing  historical  exposition.  Sir  Isaac  Newton,  1665, 
"  found  the  Proportions  of  the  Increments  of  inde- 
terminate Quantities.  These  Increments  or  Aug- 
menta  Momentanea  he  called  Moments,  which  others 
called  Particles,  infinitely  small  Parts,  and  Indi- 
visibles  ;  and  the  Velocities  by  which  the  Quantities 
increased  he  called  Motions,  Velocities  of  Increase, 
and  Fluxions.  He  considered  Quantities  not  as 
composed  of  Indivisibles,  but  as  generated  by  locai 
Motion,  after  the  manner  of  the  Ancients  .  .  .  and 
represented  such  Moments  [of  Time]  by  the  Letter  0, 

^  Republick  of  Letters,  Art.  XI,  pp.  223-235,  1736. 


TEXT-BOOKS,  1736-1741  I55 

or  by  any  other  Mark  drawn  into  an  Unit  "  (p.  228). 
'  '  Fluxions  are  not  Moments,  but  finite  Quantities 
of  another  kind."  *'When  Mr.  Newton  is  demon- 
strating  any  Proposition,  he  considers  the  Moments 
of  Time  in  the  Sense  of  the  Vulgar,  as  indefinitely 
small,  but  not  infinitely  so  ;  and  by  that  means 
performs  the  whole  work,  in  finite  Figures,  by  the 
Geometry  of  Euclid  and  Apollonius,  exactly  without 
any  Approximation  :  and  when  he  has  brought  the 
work  to  an  Equation,  and  reduced  the  Equation  to 
the  simplest  Form,  he  supposes  the  Moments  to 
decrease  and  vanish  ;  and  from  the  terms  which 
remain  he  deduces  the  Demonstration.  But  when 
he  is  only  investigating  any  Truth,  or  the  Solution 
of  any  Problem,  he  supposes  the  Moment  of  Time 
to  be  infinitely  little,  in  the  Sense  of  Philosophers, 
and  Works  in  Figures  infinitely  small." 

James  Hodgson,    1736 

152.  James  Hodgson,  a  mathematical  teacher  and 
writer,  and  a  fellow  of  the  Royal  Society  of  London, 
is  the  author  of  a  book,   The  Doctrine  of  Fluxions.^ 

Hodgson  says  in  his  Introduction  that  *'  it  is  now 
some  years  since  the  greatest  Part  of  this  Book  was 
prepared  for  the  Press."  There  is  no  direct  refer- 
ence  in  the  book  to  the  Analyst  controversy,  but 
the  declaration  is  made  that  the  principles  upon 
which    fluxions    rest    need    "  fear    no    Opposition." 

^  The  Doctrine  of  Fluxions,  fotinded  on  Sir  Isaac  Newton  s  Afethod, 
Piiblished  by  Himself  in  his  Tract  upon  the  Quadrature  of  Curves^ 
By  James  Hodgson,  London,  MDCCXXXVI. 


156  LIMITS    AND   FLUXIONS 

Hodgson  also  says  in  his  Introduction  that  most 
books  on  fluxions  that  bave  hitherto  appeared 
proceeded  on  the  same  principles  as  the  Difìferential 
Calculus,  so  that  "  by  calling  a  Differential  a 
Fluxion^  and  a  second  Differential  a  second  Fluxion, 
etc. ,  they  bave  .  .  .  confusedly  jumbled  the  Methods 
together,"  although  the  principles  are  really  "very 
difìferent.  "  '  '  The  Differential  Method  teaches  us 
to  consider  Magnitudes  as  made  up  of  an  infinite 
Number  of  very  small  constituent  Parts  put 
together  ;  whereas  the  Fluxionary  Method  teaches 
US  to  consider  Magnitudes  as  generated  by  Motion 
...  ;  so  that  to  cali  a  Diffei'ential  a  Fluxion,  or  a 
Fluxion  a  Differential  is  an  Abuse  of  Terms. "  In 
the  method  of  fluxions,  "  Quantities  are  rejected, 
because  they  really  vanish  "  ;  in  the  differential 
method  they  are  rejected  '*  because  they  are  in- 
finitely  small."  Hodgson  adds  that  he  always  used 
the  differential  method  '''till  I  became  acquainted 
with  the  Fluxionary  Method."  He  considers  fluxions 
of  quantities  (p.  50)  '*in  the  first  Ratio  of  their 
nascent  Auginents,  or  in  the  last  ratio  of  their 
evanescent  Decrements,'"  and  gives  an  able  and  faithful 
exposition  of  Newton's  ideas  as  found  in  his 
Quadrature  of  Curves.  He  cannot  think  ''there  is 
any  more  difificulty  in  conceiving  or  forming  an 
adequate  Notion  of  a  nascent  or  evanescent  Quantity, 
than  there  is  of  a  Mathematica!  Point  "  (p.  xi).  In 
explaining  the  derivation  of  the  fluxion  of  the 
product  xy^z  he  apparently  permits  (p.  xv)  the 
small  quantity  ^  to  "  vanish,"  and  thereupon  divides 


TEXT-BOOKS,  ly 7,6-1741  157 

both  sides  of  the  equation  Xfo-]-fXo  =  èo  by  0. 
However,  in  the  exposition  given  on  p.  50  he  is 
more  careful  and  divides  by  0  while  0  is  an  incre- 
ment,  and  obtd.ìns  f±-\-±y-\-J^:i;o  =  v.  Then  he  says  : 
"  Imagine  the  Quantity  0  to  be  infinitely  dimin- 
ished,  or,  which  is  the  same  thing,  the  Quantity 
xy  to  return  back  again  into  its  arising  State  ;  then 
the  Quantity  ±yo,  in  this  Case,  into  which  0  is  multi- 
pHed,  will  vanish  ;  whence  we  shall  have  xy-\-yji'=v 
for  the  Fluxion  of  the  Quantity  proposed.  "  Hodgson 
foUows  Newton  closely  and  permits  the  variable  to 
reach  its  limit. 

Thomas  Bayes,    1736 

153.  An  anonymous  pamphlet  of  50  pages,  on 
the  Doctrine  of  Fluxions,^  has  been  ascribed  to  Rev. 
Thomas  Bayes.  This  author  contributed  in  1763  to 
the  Philosophical  Transactions  a  meritorious  article 
on  the  doctrine  of  chances. 

The  pamphlet  of  1736  represents  a  careful  effort 
to  present  an  unobjectionable  foundation  of  fluxions. 
*'The  fluxion  of  a  flowing  quantity  is  its  rate  or 
swiftness  of  increase  or  decrease.  "  Let  «,  b,x^  and 
y  be  flowing  quantities,  let  A  and  B  be  permanent 
quantities  \\{  a\  b  =  h.'^x  :  B=Fjj/,  during  any  time  T, 
and  at  the  end  of  that  Time,  a^  b,  x^  y  ali  vanish  ; 
then  .  .  .  the  ratio  of  A  to  B  is  the  last  ratio  of  the 
vanishing  quantities  a  and  b  (p.  13).  This  definition 
is  *'  in  efifect  the  same"  as  that  given  by  Newton. 

^  Introduction  to  Doctrine  of  Fluxions  and  Defence  of  the  Mathe- 
maticiaus  agaitist  .   .   .  t/ie  Ana/j'st,  1736. 


158  LIMITS    AND   FLUXIONS 

The  author  speaks  of  *'that  most  accurate  defìnì- 
tion  of  the  ultimate  ratio's  of  vanìshing  quantities  ; 
which  we  have  at  the  latter  end  of  Sch,  Lemma  XI 
Princip.  [see  our  §§  10-15],  ^^^  which  is  so  plain, 
that  I  wonder  how  our  author  [Berkeley]  could  help 
understanding  it  ;  which  had  he  done,  I  am  apt  to 
think  that  ali  his  Analyst  says  concerning  the  pro- 
portion  of  quantities  vanishing  with  the  quantities 
themselves,  had  never  been  heard  :  For  according 
to  this  definition,  we  are  not  obliged  to  consider  the 
last  ratio  as  ever  subsisting  between  the  vanishing 
quantities  themselves.  But  between  other  quantities 
it  may  subsist,  not  only  after  the  vanishing 
quantities  are  quite  destroyed,  but  before  when 
they  are  as  large  as  you  please.  And  the  reason 
why  we  consider  quantities  as  decreasing  continually 
till  they  vanish,  is  not  in  order  to  make,  but  to  find 
out,  this  last  ratio.  Sir  Isaac  Newton  does  not 
indeed  say  that  this  last  ratio  is  the  ratio  with  which 
the  quantities  themselves  vanish  ;  but  whether  he 
herein  speaks  with  the  utmost  propriety  or  not,  is 
a  mere  nicety  on  which  nothing  at  ali  depends  " 
(p.   16,  note). 

Velocity  "  signifies  the  degree  of  quickness  with 
which  a  body  changes  its  situation  in  respect  to 
space";  the  fluxion  of  a  quantity  ''signifies  the 
degree  of  quickness  with  which  the  quantity  changes 
its  magnitude.  "  "And  when  our  author  asserts,  that 
in  order  to  conceive  of  a  second  fluxion,  we  must 
conceive  of  a  velocity  of  velocity,  and  that  this  is 
nonsense  ;    he    plainly    appeals    to    the    sound    and 


TEXT-BOOKS,  17  36-1 741  159 

not  the  sense  of  words  .  .  .  if  .  .  .  you  make  it 
synonymous  to  the  word  Fluxion,  then  the  velocity 
of  velocity  .  .  .  is  nothing  but  plain  common 
sense"  (p.  19).  Moments  are  not  used  by  the 
author.  The  author  says  that,  were  he  to  write 
a  treatise  on  fluxions,  **in  order  to  understand 
equations  where  Fluxions  of  different  orders  are 
jumbled  together  ;  it  would  be  convenient  to  re- 
present  ali  Fluxions  not  as  before,  but  as  quantities 
of  the  same  kind  with  their  Fluents.  .  .  .  The 
Fluxion  of  a  quantity  anyhow  flowing  at  any  given 
instant  is  a  quantity  found  out  by  taking  it  to  the 
Fluxion  of  an  uniformly  flowing  quantity  in  the 
ultimate  proportion  of  those  synchronal  changes 
which  then  vanish  "  (pp.  34,  35).  The  variables 
X    and   x"^   have    the    synchronal    augments    0    and 

nox''~^-\-^ — —^o^x"~^-\-,    etc. ,    which    are    to    one 

2 

another   as    i  :  nx''~'^ -\-- ^-ox""'^-]-,   etc.      *'Let 

2 

now  these   arguments   vanish,    and  their  last  ratio 

will   be    I  :  nx^'K"      *'This   our  author   says  is   no 

fair    and    conclusive    reasoning,    because    when    we 

suppose  the  *  increments  to  vanish,  we  must  suppose 

their  proportions,  their  expressions,  and  everything 

else  derived  from  the  supposition  of  their  existence 

to  vanish  with  them.  '     To  this   I   answer,  that  our 

author   himself  must  needs  know  thus    much,   viz. 

That  the  lesser  the  increment  0  is  taken,  the  nearer 

the  proportion  of  the  increments  of  x  and  x""  will 

arrive  to  that  of  i   to  nx"'"^,  and  that  by  supposing 


i6o  LIMITS    AND   FLUXIONS 

the  increment  o  continually  to  decrease,  the  ratio 
of  these  synchronal  increments  may  be  made  to 
approach  to  it  nearer  than  by  any  assignable 
difìference,  and  can  never  come  up  with.  it  before 
the  time  when  the  increments  themselves  vanish. 
.  .  .  For  tho',  strictly  speaking,  it  should  be 
allowed  that  there  is  no  last  proportion  of  vanishing 
quantities,  yet  on  this  account  no  fair  and  candid 
reader  would  find  fault  with  Sir  Isaac  Newton,  for 
he  has  so  plainly  described  the  proportion  he  calls 
by  this  name,  as  sufficiently  to  distinguish  it  from 
any  other  whatsoever  :  So  that  the  amount  of  ali 
objections  against  the  justice  of  this  method  in 
finding  out  the  last  proportion  of  vanishing 
quantities  can  arise  to  little  more  than  this,  that 
he  has  no  right  to  cali  the  proportions  he  finds  out 
according  to  this  method  by  that  name,  which  sure 
must  be  egregious  trifling.  However,  as  on  this 
head  our  author  seems  to  talk  with  more  than  usuai 
confidence  of  the  advantage  he  has  over  his  oppo- 
nents,  and  gives  us  what  he  says  is  the  amount  of 
Sir  Isaac's  reasoning,  in  a  truly  ridiculous  light,  it 
will  be  proper  to  see  on  whom  the  laugh  ought  to 
fall,  for  I  am  sure  somebody  must  bere  appear 
strangely  ridiculous,  ...  I  readily  allow  whatever 
consequence  he  is  pleased  to  draw  from  it,  if  it 
appears  that  Sir  Isaac,  in  order  to  find  the  last 
ratios  proposed  was  obliged  to  make  two  incon- 
sistent  suppositions.  To  confute  which  nothing 
more  need  be  said  than  barely  to  relate  the  sup- 
positions he  did  make. 


TEXT-BOOKS,    1736-1741  161 

*'  I.  Then  he  supposes  that  x  by  increasing  bè- 
comes;ir  +  ^,  and  from  hence  he  deduces  the  relation 
of  the  increment  of  x  and  ;r". 

'*2.  Again,  in  order  to  find  the  last  ratio  of  the 
increments  vanishing,  he  supposes  0  to  decrease  till 
it  vanishes,  or  becomes  equal  to  nothing.  .  .  . 
These  are  evidently  no  more  inconsistent  and  con- 
tradictory,  than  to  suppose  a  man  should  first  go 
up  stairs,  and  then  come  down  again.  To  suppose 
the  increment  to  be  something  and  nothing  at  the 
same  time,  is  contradictory  ;  but  to  suppose  them 
first  to  exist,  and  then  to  vanish,  is  perfectly  con- 
sistent  ;  nor  will  the  consequences  drawn  from  the 
supposition  of  their  prior  existence,  if  just,  be  any 
ways  affected  by  the  supposition  of  their  subsequent 
vanishing,  because  the  truth  of  the  latter  supposition 
no  ways  would  have  been  an  inconsistency  ;  but  to 
suppose  them  first  unequal,  and  afterwards  to  become 
equal,  has  not  the  shadow  of  difficulty  in  it.  .  .  . 
must  confess  there  seems  to  be  some  objection  against 
considering  quantities  as  generated  from  moments. 
What  moments,  what  \)i\^  principia  janijam  nascentia 
finitaruiìi  qziantitatum,  are  in  themselves,  I  own,  I 
don't  understand.  I  can't,  I  am  sure,  easily  con- 
ceive  what  a  quantity  is  before  it  comes  to  be  of 
some  bigness  or  other  ;  and  therefore  moments 
considered  as  parts  of  the  quantities  whose  moments 
they  are,  or  as  really  fixed  and  determinate  quanti- 
ties of  any  kind,  are  beyond  my  comprehension,  nor 
do  I  indeed  think  that  Sir  Isaac  Newton  himself  did 
thus  consider  them  "  (pp.  35-41). 

Il 


i62  LIMITS    AND   FLUXIONS 

John  Muller,   1736 

154.  John  Muller,  a  German  by  birth,  dates  bis 
Mathematical  Treatise^^  173^,  from  the  Tower  of 
London,  and  dedicates  it  to  the  master-general  of 
the  ordnance.  He  was  appointed  in  1741  head- 
master  of  the  Royal  Military  Academy,  Woolwich. 
He  was  *  '  the  scholastic  father  of  ali  the  great 
engìneers  this  country  employed  for  forty  years.  " 

The  author's  method  of  explaining  fluxions  is 
somewhat  unique.  *'  I  make  no  use  of  infinitely 
small  quantities  nor  of  nascent  or  evanescent 
velocities  ;  and  yet  I  think  to  bave  explained  those 
Principles,  so  that  any  Person  of  a  moderate 
capacity  .  .  .  may  be  fully  convinced  of  the  Truth 
thereof"  (Preface).  He  begins  his  conic  sections 
with  the  postulate:  ''Grant  that  two  infinite 
quantities,  differing  from  each  other  by  a  finite 
quantity,  may  be  esteemed  equal.  "  He  then 
explains  that  this  postulate  '*is  bere  of  use  only 
to  shew  the  connection  of  the  Conic-Sections,"  and 
hastens  to  assure  the  reader  that  ''whenever  we 
make  use  of  it  in  the  demonstration  of  any  Proposi- 
tion,  we  shall  give  always  another  Demonstration 
independent  on  it." 

In  the  Republick  of  Lctters,  June,  1736,  occurs  the 
foUowing  comment  : 

"  He  introduces  this  [Conic-Sections]  by  a  Postu- 
latum  that  sounds  very  absurdly  to  those  that  are 

^  A  Mathematical  Treatise  :  Containing  a  System  of  Conic-Sections  ; 
with  the  Doctrine  of  Fluxions  and  Fluents,  Applied  to  various  Subjects. 
By  John  Muller.     London,  1736. 


TEXT-BOOKS,    1736-1741  163 

not  vers'd  in  mathematica!  Speculations.  'Grant,' 
says  he,  '  that  two  infinite  Quantities,  differing  from 
one  another  by  a  finite  Quantity,  may  be  esteemed 
equal.  '  Such  would  imagine  that  there  could  not 
be  two  infinite  Quantities  ;  or  that  if  there  could, 
they  must  necessarily  be  absolutely  and  not  only 
reputedly  equal.  But  however  Hobbes  or  Berkeley 
may  talk  of  geometrica!  Fallacies,  or  these  unex- 
perienced  People  think,  the  Adepts  in  this  Science 
very  well  know,  that  more  infinite  Quantities  than 
two  are  possible,  and  that  one  Quantity  may  be  in- 
finitely  greater  than  an  infinite  one,  and  yet  be  itself 
infinitely  less  than  a  third.  But  enough  of  these 
Ludibria  Scientiae,  that  I  may  inform  the  Publick  of 
the  more  useful  Theorems  ..."  (pp.  422,  423). 

Muller  considers  in  his  text  a  curve  generated 
by  a  point  "  urged  by  two  powers  acting  in  two 
dififerent  directions,  the  one  parallel  to  the  Abscisses 
and  the  other  parallel  to  the  Ordinates.  I  prove 
from  thence,  that  if  this  point  (when  arrived  at  a 
given  place)  did  continue  to  move  with  the  velocity 
it  has  there,  it  would  proceed  in  a  right  line  touch- 
ing  the  Curve  in  that  place  ...  So  that  the  three 
Directions  being  known  in  each  place,  the  propor- 
tion  between  the  velocities  of  the  urging  powers  is 
likewise  known."  Fluxions  are  defined  as  velocities. 
To  find  the  fluxion  of  j^^^  he  puts  y'^=x\  the  sub- 
tangent  of  the  parabola  is  2y'^  Since  the  subtangent 
is  to  the  ordinate  as  the  velocities  along  the  abscissa 
and  ordinate,  he  has  2y^  -.  y  \  \x  \  y,  or  ,r=2yy,  and 
2yy  is  the  required  fluxion.      Similarly,  to  find  the 


64  LIMITS   AND  FLUXIONS 


fluxion  of  j|/^,  let  x  =y^.  Take  u~z^^  the  u—x=z  —y 
y.zz-\-2y-\-yy^  or  z^y  :  u—x=i  :  zz-\-zy-\-yy.  li 
now  j/  and  z  approach  continually  until  they  coincide 
with  an  intermediate  ordinate,  then  z=y  and  the 
chord  through  the  extremities  of  the  ordinatesi  and 
z  will  likewise  coincide  with  the  tangent.  Therefore, 
the  ordinate  is  to  the  subtangent  as  i  is  to  3j/^. 
Hence  the  proportion  i  :  3XF=>'  :  i',  or  i'=3>x^,  the 
fluxion  required.  The  same  argument  is  applied  to 
y^".  In  these  demonstrations  appeal  is  made  to  a 
geometrie  figure,  and  no  attention  is  directed  to  the 
ratio  z—y  \u—x  for  the  difficult  case  when  j  =  5. 
The  author  remarks  that  "  though  we  commonly 
say  that  .  .  .  mjy'*''^  is  the  Fluxion  ofy  ;  yet  that 
expression  is  not  sufficiently  accurate  :  Therefore, 
the  sense  in  which  we  desire  to  be  understood  is, 
that  I  :  7ny"^~^  :  :  y  :  7nyy'"'^,  that  is,  unity  is  to 
uiy"'''^,  or  j>  is  to  myy'"~^^  as  the  fluxion  or  velocity 
with  which  y  is  generated,  is  to  the  fluxion,  or 
contemporary  velocity  with  which  j/"'  is  generated, 
and  so  for  the  rest  "  (p.  79).  Thus,  the  emphasis. 
is  placed  upon  the  ratios  of  velocities. 

Anonymous   Trans lation  ^  of  Neiuton's 
'  '  Method  of  Fluxio  ns,''   1737 

i^^a.  Colson's  translation  from  the  Latin  ol' 
Newton's  Method  of  Fluxions,  published  in  1736,, 
was  foUowed  in  1737  by  a  second  translation,  which 

^  A  Treatise  of  the  Method  of  Fluxions  and  Infinite  Series^  With  its 
Application  to  the  Geonictry  of  Curve  Lines.  By  Sir  Isaac  Newton,  Kt, 
Translatcdfrom  the  Latin  Orioinal  not  yet  published.  Designed  by  the^ 
Author  for  the  Use  of  Lear  neri.     London,  MDCCXXXVII. 


TEXT-BOOKS,    1736-1741  165 

was  anonymous.  In  it  no  mention  is  macie  of 
Colson's  edition.  The  anonymous  translator  says 
in  the  preface  :  '  *  We  have  reason  to  believe  that 
what  is  here  delivered,  is  wrought  up  to  that  Fer- 
fection  in  which  Sir  Isaac  himself  had  once  intended 
to  give  it  to  the  Publick.  The  ingenious  Dr. 
Pemberton  has  acquainted  us  that  he  had  once  pre- 
vailed  upon  him  to  complete  his  Design  and  let 
it  come  abroad.  But  as  Sir  Isaac's  Death  un- 
happily  put  a  stop  to  that  Undertaking,  I  shall 
esteem  it  none  of  the  least  Advantages  of  the 
present  Publication,  if  it  may  prove  a  means  of 
exciting  that  Honourable  Gentleman,  who  is 
possessed  of  his  Papers,  to  think  of  communicating 
them  to  some  able  Hand  ;  that  so  the  Piece  may  at 
last  come  out  perfect  and  entire."  As  remarked 
by  G.  J.  Gray,^  the  two  translations  were  made 
*'from  copies  of  the  same  manuscript,"  and  differ 
from  each  other  only  "in  the  mode  of  expressing 
the  work  in  English.  " 

James  Smith ^    ^717 

155.  In  his  New  Treatise  of  Fluxions,^  Smith 
says  (Preface)  :  ''  What  I  cali  here  the  New  Method, 
and    the   Six    Propositions    immediately   following, 

^  A  Bibliography  of  the  Works  of  Sir  Isaac  Newton.  By  George 
J.  Gray.     Second  edition,  Cambridge,  1907,  p.  47. 

2  A  New  Treatise  of  Fhixions,  containitig,  I.  The  Elements  of 
Fluxions,  demonstrated  in  Two  easy  Propositions,  without  first  or  last 
Ratios,  IL  A  Treatise  of  Nascent  and  Evanescent  Quantities,  first  and 
last  Ratios,  III.  Sir  Isaac  Newton  s  Detnonsiration  of  the  Tluxions 
enlarged  and  illustrated :  IV.  Ansiucrs  to  the  Principal  Ohjections  in 
the  Analyst.     By  James  Smith,  A.M.,  London,  1737. 


i66  LIMITS    AND   FLUXIONS 

are  entirely  New  .  .  .  Our  common  Definition  of 
Motion,  trans  latto  corporis  de  loco  in  locum  is 
certainly  imperfect,  and  1  am  inclined  to  think, 
that  Aristotle's  old  exploded  Definition  of  Motion 
will,  some  time  or  other,  come  into  Vogue  again. 
Actus  entis  in  potentia^  cpiatcnus  in  potcìitia  est. 
Motion  is  an  Efifect,  and  every  Effect  has  a  coin- 
staneous  Existence  with  the  Action  by  which  it  is 
produced." 

The  definitions  with  which  Smith  starts  out  are 
not  very  reassuring.  "The  fluxion  of  a  surface  is 
the  Velocity  of  the  generating  Line."  "The 
velocity  of  a  generating  Line  is  the  Sum  of  the 
Velocities  of  ali  the  Points  of  that  Line,  whether 
these  Points  move  with  equal,  or  unequal  Veloci- 
ties. "  The  rectangle  xy  *  *  flows  or  increases  by  the 
flowing  of  both  its  contiguous  Sides  "  together  ; 
but  it  "flows  into  Length  "  by  the  velocity  ja-,  and 
**it  flows  into  Breadth  at  the  very  same  Instant 
of  Time"  by  the  velocity  xy.  "  Therefore  the 
Velocity  with  which  it  flows  into  Length  and 
Breadth  is  the  Sum  of  the  synchronic  Velocities," 
xy-\-yx. 

Nor  is  the  second  topic  displayed  with  illumina- 
tion.  "A  nascent  Ouantity  is  a  Quantity  in  the 
Instant  of  its  commencing  to  exist."  Similar  to 
this  is  the  definition  of  "  evanescent  Quantity,"  as 
are  also  the  definitions  of  first  and  last  ratios. 
Interesting  is  the  following  proof  that  if  "  two 
Quantities  begin  and  cease  to  exist  in  any  finite 
Time  T,   .   .    .   they  have  a  first  and  a  last   Ratio," 


TEXT-BOOKS,    1736-1741  167 

for,  '*if  they  have  not  a  first  Ratio,  they  have  not 
a  second  nor  a  third  Ratio,  etc.  Therefore  they 
have  no  Ratio  in  the  Time  T  ;  but  in  the  Time  T 
they  are  Ouantities,  "  and  *  *  two  quantities  of  the  same 
kind,  as  soon  or  so  long  as  they  have  any  Quantity, 
Being  or  Existence  {i.e.  are  not  absolutely  nothing], 
have  a  Ratio  the  one  to  the  other,"  that  is,  *'they 
have  a  Ratio  and  they  have  not  a  Ratio  in  the 
Time  T,  which  is  absurd."  Smith  argues  also  that 
since  two  quantities  ''cannot  be  in  their  first  Ratio, 
neither  before  nor  after  the  Beginning  of  the  Time 
T,  they  must  have  been  in  their  first  Ratio  at  the 
very  Beginning  of  the  Time  T,  just  as  they  began 
to  exist."  Near  the  dose  of  this  part  of  his  hook, 
Smith  reveals  some  of  the  subtleties  of  his  topic 
by  stating  an  **Objection"  and  the  ''Answer"  to 
it.  The  Objection  :  *'Nascent  and  evanescent 
Quantities  are  Something  or  Nothing  ;  for,  Inter  ens 
et  non-ens  non  daiur  medium.  If  Something,  then 
the  Ratio  of  evanescent  Ouantities  is  the  same 
with  the  Ratio  before  they  were  evanescent, 
or  when  they  had  any  finite  Magnitude.  .  .  . 
If  they  are  mere  Nothing,  or  Non-quanta  ;  then 
B^/E^  =  0/0  =  0;  .  .  .  which  is  absurd."  In  the 
*'Answer"  Smith  says  :  "Evanescent  Quantities 
are  really  nothing,  or  Non-quanta  ;  for  it  is  evident 
.  .  .  that  upon  ^'s  coinciding  with  B,  and  /s  co- 
inciding  with  E,  the  Increments  B/;  and  E^  are 
annihilated,  and  evanescent  Quantities  are  never 
accurately  evanescent,  but  upon  this  or  the  like 
Coincidence.      And  yet  it  does  not  follow  that  their 


i68  LIMITS   AND  FLUXIONS 

last  Ratio,  or  the  Ratio  they  nihilesce  with,  is 
Nothìng.  For  B^  /  E^  is  neither  B/;  nor  E^,  nor 
B^  and  E^,  but  a  Mark  or  Expression  of  their 
Ratio,  which  may  be  expressed  as  well  by  any 
other  Character.  .  .  .  The  Increments  are  indeed 
annihilated  and  gone,  but  their  last  Ratio  remains, 
and  is  as  real  as  any  Ratio  they  ever  had  "  ;  .  .  . 
they  have  as  real  a  Ratio  at  the  last  Instant  of 
their  Existence  ;  that  is,  when  they  are  ceasing  to 
be  Something,  and  commencing  to  be  Nothing,  as 
they  had  at  any  instant  preceding  the  last  Instant 
of  their  Existence."  .  .  .  ''There  is,  sometimes, 
something  very  strange  in  the  Nature  of  these 
evanescing  Augments,  and  it  is  literally  true  of 
them,  what  Juvenal  figuratively  says  of  Man. 

— Mors  sola  fatetur^ 
Quantula  sunt  honiinum  corpuscula — 

We  know  nothing  of  them  till  they  be  dead  and 
gone." 

Of  Part  III,  in  which  Smith  '*  demonstrates  " 
Newton's  Method  of  Fluxions,  we  quote  only  the 
last  sentence  :  *'  I  have  made  use  of  infinitely  little 
Quantities,  and  of  a  second  Point  as  being  next  to 
a  first  Point  ;  but  this  was  only  for  Illustration  sake. 
There  is  not  the  least  Occasion  for  any  of  these 
Notions  in  the  Demonstration.  " 

In  the  last  part  of  Smith's  hook,  Berkeley's  con- 
tention,  '*  No  just  Conclusion  can  be  drawn  from 
two  contrary  Suppositions,"  is  answered  by  the 
statement,  ''This  is  certainly  true,  Ì7i  sensu  com- 
posito^ but  in  sensu  diviso  is  intirely  false." 


TEXT-BOOKS,    1736-1741  169 

We  are  tempted  to  make  the  remark  that  in 
1737  Smith  left  the  subject  even  more  mysterious 
than  he  found  it. 

Thomas  Siinpson^  ^72)7 

156.  Thomas  Simpson,  the  son  of  a  vveaver,  was 
a  self-taught  mathematician,  and  acquired  a  know- 
ledge  of  fluxions  through  Stone's  translation  of  De 
L'Hospital's  A^ialysc  des  infinimcnt pctits.  Simpson 
was  a  mathematician  of  marked  power,  and  influenced 
considerably  the  teaching  of  mathematics  in  England. 
In  1737  he  broughtout  his  New  Treatise  of  Fluxions  ,'^ 
which  contains  some  novel  features. 

**  The  Fluxions  of  variable  Ouantities  are  always 
measured  by  their  Relation  to  each  other  ;  and  are 
ever  expressed  by  the  finite  Spaces  that  would  be 
uniformly  described  in  equal  Times,  with  the  Veloci- 
ties  by  which  those  Quantities  are  generated." 

He  finds  it  easy  to  show  that  the  fluxion  of  a 
rectangular  area  of  Constant  height  and  uniformly 
variable  base  is  as  the  height  drawn  into  the 
velocity  with  which  the  base  changes  ;  also  that 
the  fluxion  of  a  curvilinear  area  generated  by  an 
abscissa  moving  with  uniform  velocity  is  at  a  given 
point,  as  the  ordinary  y  for  this  point,  multiplied 
by  that  velocity.  This  last  result  is  applied  to 
finding  the  fluxion  of  ,tT. 

Avoiding  infinitely  small  quantities,  Simpson  finds 

^  A  Ne7v  Treatise  of  Fluxions:  whercin  the  direct  and  ifiverse 
Method  are  demonstrated  after  a  new,  clear  and  concise  Manner,  with 
their  Application  to  Physics  and  Astronomy,  By  Thomas  Simpson, 
London,  1737. 


I70  LIMITS    AND   FLUXIONS 

the  ratio  of  the  fluxions  of  x  and  x^  thus  :  Let  the 
points  iìi  and  n  move  so  that  the  distance  h  de- 
scribed  by  n  shall  always  equal  the  square  of  the 
distance  g  described  by  ni  in  the  same  tinne.  Then 
(AR)2  =  CS,  (AR-Rr)2=:Cj,  and  jS  =  2AR  x  Rr- 
(Rr)=^.  But  jS  is  described  with  accelerated  velocity 
when  ;;/  moves  uniformly,  hence  ^S  will  be  'Mess 
than  that  which  would  be  uniformly  described  in 
the  same  time  with  the  Velocity  at  the  point  S, 
and   greater   than   that   which   would   be   described 

with  the  Velocity  at  the 
,g  point  s  ;  and  therefore 
must  be  equal  to  the  Dis- 
tance that  would  be  uni- 
formly described  with  the 
Velocity  at  another  point 
e  posited  somewhere  be- 
tween  S  and  s,  in  the  same 
Time  that  the  other  point  m  is  moving  over  the 
Distance  rR  ;  therefore  rR  :  2ARxRr-(Rr)2  :  : 
^:^(2AR  — Rr),  the  Distance  that*  would  be  de- 
scribed with  the  Velocity  of  n^  at  the  point  e,  in  the 
same  Time  that  in  is  moving  over  the  Distance  g  : 
Now  therefore  when  the  points  r  and  s  coincide 
with  R  and  S,  then  will  e  coincide  with  S  ;  .  .  .  and 
consequently  (2AR  — Rr)^^  will  then  .  .  .  become 
2ARx^,  equal  to  h  the  required  Distance."  The 
criticai  part  of  this  proof  is  '  '  when  the  points  r  and  s 
coincide  with  R  and  S,  then  will  e  coincide  with  S." 
A  modification  of  this  proof  is  applied  to  x"". 

Simpson's  text  marks  a  departure  from   Newton 


rt . 

r 

R 

H' 

m 

m 

m  ' 

r , 

S 

e  S 

o* 

n 

n 

n 

9 

-/? 

TEXT-BOOKS,    1736-1741  171 

in  the  definition  of  fluxion.  Newton  makes  it  a 
velocity,  Simpson  makes  it  a  finite  distance.  On 
the  necessity  and  wisdom  of  this  change  there  can 
readily  be  difference  of  opinion.  But  there  can  be 
no  denying  that  Simpson  developed  his  theory  of 
fluxions  in  a  manner  almost,  though  not  entirely, 
free  from  the  objections  against  fluxions  that  had 
been  advanced  by  Berkeley  ;  infinitely  small  quan- 
tities  are  nowhere  used.  A  short  but  appreciative 
review  of  this  text  appeared  in  The  Works  of  the 
Learned  for  July,    1737. 

Benjamin  AI  art  in,    1739,    1759 

157.  Benjamin  Martin  was  a  mathematician,  an 
optical  instrument  maker,  and  a  general  compiler. 
He  was  a  self-educated  man,  and  at  one  time  taught 
reading,  writing,  and  arithmetic.  His  exposition  of 
fluxions,  as  found  in  his  Eleinents  of  ali  Geonietry  ^ 
and  in  a  later  work,  is  below  the  standard  usually 
reached  by  him  in  mathematica!  writing. 

This  book,  intended  as  an  introduction  to  modem 
mathematics,  contains  in  an  Appendix  an  epitome 
of  the  doctrine  of  fluxions.  '*  Since  Fluxions  are 
the  very  small  Increnients  and  Decrenients  of  the 
Flowing  Ouantities,  or  the  Velocities  of  the  Motions 
whereby  they  increase  or  decrease,  'tis  plain  that 
those  Fluxions,  or  Velocities,  themselves  may  be 
consider'd  as  Flowing  Quantities,  and  their  Fluxions 
are  call'd  Second   Fluxions  ..."      It  would  seem 

1  nANrEi2METPIA  ;  or  ihe  Elements  of  ali  Ceometry.    By  B.  Martin, 
London,  M.DCC.XXXIX. 


1/2  LIMITS    AND  FLUXIONS 

that  in  this  statement  a  fluxion  is  "  very  small  "  and 
at  the  same  time  a  *'velocity."  A  little  later  the 
author  refers  to  fluxions  as  ''in  the  first  Ratio  of 
Augmenta  nascentia,  "  Evidently,  in  this  Appendix, 
covering  twelve  pages,  the  author  has  not  succeeded 
in  presenting  a  consistent  theory  of  fluxions. 

A  fuller  exposition  was  given  twenty  years  later 
in  the  System  of  Mathcmatical  InstitutionSy  agrecable 
to  the  Present  State  of  the  Newtonian  Mathesis^ 
by  Benjamin  Martin,  voi.  i,  London,  MDCCLIX. 
The  theory  is  stili  confusing.  "  Indefinitely  small 
Spaces  "  (p.  362)  are  represented  by  x  and  j>,  which 
are  called  the  fluxions  oi  x  andjj^,  and  said  to  repre- 
sent  the  velocities  of  moving  points.  Newton  is 
reported  to  have  at  first  delivered  the  idea  of  what 
Martin  calls  a  fluxion,  under  the  name  of  momentum^ 
"  a  Term  used  in  Mechanics  to  denote  the  Quantity 
of  Motion  generated  by  a  given  Quantity  of  Matter 
(A),  and  the  Velocity  {a)  with  which  it  moved  con- 
jointly.  This  Momentum  therefore  was  properly 
represented  by  (A^).  .  .  .  But  instead  of  this 
mechanical  Notation,  we  now  use  xx  and  yy  for  the 
Momenta^  or  Fluxions.  ..."  It  is  seldom  that 
one  encounters  a  more  grotesque  conglomeration  of 
unrelated  ideas  than  is  presented  here.  Martin 
gives  John  Rowe's  mode  of  deriving  the  fluxions 
of  xy  and  xyz. 

An  Anonynious    Text,    1741 

I  58.  An  Explanation  of  Fluxions  in  a  Short  Essay 
on  the  Theo7y.     London:  Printed  for  W.  Innys,  at  the 


TEXT-BOOKS,    1*736-1741  173 

West-End  of  St.  Paul's,  MDCCXLI.  This  anony- 
mous  publication  of  16  pages  was  reprinted  in  1809 
in  the  fourth  edition  of  John  Rowe's  Doctrine  of 
Fluxions  ;  it  constitutes  a  real  contribution  to  the 
logie  of  fluxions.  The  pamphlet  is  offered  "  as  an 
Explanation  of  the  Doctrine  itself,  and  not  of  Sir 
Isaac's  Manner  of  delivering  it."  '*About  that," 
he  says,  "  I  don't  mean,  nor  pretend  to  take  a  Part 
in  any  Controversy. "  He  defines  fluxions  thus  : 
**The  word  Fluxion  properly  apply'd  always  sup- 
poses  the  Generation  of  some  Quantity  (term'd 
Fluent  or  Flowing  Quantity)  with  an  equable, 
accelerated,  or  retarded  Velocity,  and  is  itself  the 
Quantity  which  might  be  uniformly  generated,  in  a 
Constant  Portion  of  Time,  with  the  Amount  or 
Remainder  of  that  Velocity,  at  the  Instant  of  find- 
ing  such  Pluxion."  "  Hence,  it  will  appear  that 
the  first  Fluxions  of  Quantities  are  as  the  Velocities 
with  which  those  Quantities  are  increas'd  ;  that 
second  Fluxions  are  as  the  Increase  or  Decrease  of 
such  Velocities  ;  and  that  by  second,  third,  fourth, 
etc. ,  Fluxions  are  meant  Fluxions,  whose  Fluents 
are  themselves  Fluxions  to  other  proposed  Quan- 
tities ;  and  the  manner  of  considering  and  determin- 
ing  them  is  the  very  same  as  tho'  they  were  first 
Fluxions,  they  being  actually  so  to  the  Quantities 
from  which  they  are  immediately  derived  "  (p.  7). 

Then  follows  the  lemma  : 

"The  Fluxion  of  the  Area  ABC,  whether  tri- 
.angular  or  curvilinear,  is  the   Rectangle  ij." 

Suppose  B  to  move  along  hV  while  the  ordinate 


174 


LIMITS    AND   FLUXIONS 


G             C 

^ — 1 

V 

/      ^ 

f      X 

oc 

y  terminatcs  in  the  curve  AC  ;  "  And,  at  any  pro- 
peseci Position  BC,  conceive  y  to  become  Constant," 
while  B  "  moves  uniformly  any  Constant  Time,  ?//;/, 
with  the  Velocity  at  B,  over  the  Distance  x  or  BD  ; 
for  then  vigili  y  in  the  Time  mn  uniformly  generate 
the  Rectangle  ij,  which  Rectangle  is  plainly  the 
Fluxion  of  ABC  in  this  Position  {per  Definii.).'" 
Then  follows  the  illuminating  scholium  :  "  It  has 
been  commonly  objected  to  the  Accuracy  of  Fluxions, 

that  the  Trapezium  or  curvi- 
linear  Space  BC<^^D,  not  the 
Rectangle  xy^  is  the  Fluxion 
geometrically  exact.  But, 
^  this  Objection  is  built,  I 
apprehend,  upon  a  false  Idea 
of  the  Thing.  It  supposes  a 
Fluxion  a  complete  Part  of 
r  a  flowing  Quantity,  and  an 
Infinity  of  Fluxions  to  con- 
stitute  the  flowing  Quantity, 
which  are  Mistakes  {per  Definition  and  Lemma) 
.  .  .  if  i-  be  imagined  ìnfinitely  little,  an  Infinity  of 
Increments  may  constitute  the  Area  ABC.  But,  in 
Fluxions,  our  Reasoning  is  quite  differenti  a  Pluxion 
can  no  more  be  called  a  Part  of  the  Fluent,  than  an 
Efìfect  a  Part  of  the  Cause.  For  Instance;  from  the 
Fluxion  given  we  know  the  Fluent,  and  vice  versa, 
just  as  when  a  Cause  is  known  to  produce  a  certain 
Effect,  we  can  infer  the  one  from  a  Knowledge  of 
the  other.  " 

We  shall  find  that  later  this  reference  to  cause  and 


B 

FiG.  8, 


TEXT-BOOKS,    1736   1741  175 

efifect  figured  in  a  controversy  carried  on  against 
Simpson. 

As  regards  the  lemma  given  above,  we  shall  see 
that  the  same  idea  is  elaborated  in  detail  by 
Maclaurin  in  his  work  and  that  a  short  and  even 
more  convincing  statement  than  the  one  given  here 
is  found  in  the  later,  revised,  text  of  John  Rowe. 

From  the  above  lemma,  the  derivation  of  the 
fluxion  of  xy  becomes  easy  by  considering  the  rect- 
angle  ABCG  as  made  up  of  two  parts  AHCB  and 
AHCG,  and  applying  the  lemma  to  each  part. 

John  Rowe,  1741,  1757,  1767 

159.  The  first  edition  (1741)  of  John  Rowe's 
Doctrine  of  Fluxions  ^  appeared  anonymously.  A 
copy  in  the  British  Museum  has  the  following  added 
by  band  after  the  preface  :  "  This  is  the  first  edition 
of  John  Rowe's  Fluxions.  The  second  came  out 
withhis  name  in  1757  with  alterations  and  additions, 
and  the  third  came  out  in  1767  much  improved.  " 

In  the  first  edition  Rowc  begins  by  stating  his  pro- 
gramme:  ''To  render  the  Doctrine  of  Fluxions  plain 
and  easy"  by  explaining  their  nature  *'as  deliver'd 
both  by  Sir  Isaac  Newton  and  by  Leibniz."  Accord- 
ing  to  Newton,  **  Pluxion  is  the  same  as  velocity." 

''Definition  II  [Foreigners  Definition],  Quantities 
are  here  suppos'd  to  be  generated  by  a  continuai  In- 
crease,  as  before;  and  the  indefinitely  small  Particles 

^  An  Introdudion  to  the  Doctrine  of  Fluxions.  Revised  by  several 
Gentlemen  well  skiird  in  the  Mathematics.  Felicibus  inde  Ingeniis 
aperitur  IXar—Claiidian.     London,   M.DCC.XLI, 


176 


LIMITS    AND   FLUXIONS 


whereby  they  are  continually  increas'd,  are  call'd  the 
Fluxions  of  these  Quantities  "  (p.  3).  '*This  is 
the  Notion  of  Fluxions  as  deliver'd  by  Leibnitz  and 
his  Followers.  But  these  Fluxions,  we  shall,  in  the 
following  Sheets,  cali  by  the  Names  of  Moments^ 
Increments  and  Dccreinents  ;  that  is,  Moments  or 
Increments  when  the  variable  Quantities  are  increas- 
ing,  and  Decrements  when  they  are  decreasi ng  "  (p.  4). 

"As  the  Point  b  is 
continually  nearer  to  a 
Coincidence  with  the 
Tangent  TBG  the 
nearer  it  approaches  the 
Point  of  Contact  B;  so 
if  we  conceive  the  Ordi- 
nate cb  to  be  moved  on 
till  it  concides  with  CB  ; 
the  very  first  moment 
before  its  Coincidence,  the  Curve  B(^,  and  Right 
line  BG  will  be  infinitely,  or  rather  indefinitely 
near  a  Coincidence  with  each  other  ;  and  conse- 
quently,  in  that  Case,  the  Increments  B^,  and  eb 
will  come  indefinitely  near  to  measure  the  Ratio 
of  the  Fluxions  of  the  Absciss  and  Ordinate  AC, 
and  CB,  or  the  Velocities  with  which  they  flow  at 
the  Point  B  .  .  .  and  therefore  (because  when  any 
Ouantity  is  increas'd  or  decreas'd,  but  by  only 
an  infinitely  or  indefinitely  small  Particle,  that 
Ouantity  may  be  consider'd  as  remaining  the  same 
as  it  was  before  ;)  these  Increments  may  be  taken 
as  Proportional  to,  or  for  the  Fluxions  in  ali  Opera- 


TEXT-BOOKS,    1736-1741 


77 


tions  ;  and,  on  the  contrary,  the  Fluxion  for  the 
Increment  "  (pp.  5,  6).  Accordingly,  he  deduces  the 
rules  of  operation  by  the  use  of  increments,  and  in 
the  result  substitutes  the  fluxion  for  the  increment. 

In  finding  the  fluxion  of  xy  he  lets  x'  and  y'  be 
the  increments,  then  the  "  increase  in  its  nascent 
state"  is  such  that  x'y'  ''bears  no  assignable  Ratio 
to  either  x'y  or  xy  (for  as  x'y'  :  x'y  :  :y'  :  y  and  y'  by 
Supposition  is  infinitely  less  than  y,''  and  can  be 
"  expunged  or  rejected.  " 

160.  The  third  edition  (1767)  was  commented  upon 
by  J.  Stubbs,  Fellow  of  Queen's  College,  Oxford, 
as  follows  :  *'  I  received  your  valuable  present,  and 
was  much  surprised  to  find 
it  so  prodigiously  improved. 
Indeed,  it  so  much  resembles 
a  New  Work,  when  compared 
with  the  First  Edition,  that  I 
almost  wish  you  had  made 
no  mention  of  its  being  the 
Third  ;  but  left  the  two  former 
to  be  forgotten." 

The  fluxion  of  xy  is  now 
deduced  thus  :  "The  fluxion 
of  the  curvilinear  space  AEI  is  less  than  the  fluxion  of 
the  rectangle  (of  Constant  altitude)  AH  before  EH 
reaches  BC,  and  greater  after  EH  passes  BC  ;  hence 
at  BC  the  two  fluxions  are  alike  and  equal  to  yi: 
Similarly,  it  follows  that  the  rectangle  AG  (of  Con- 
stant base)  has  the  same  fluxion  xy  at   DB  as  has 

the    curvilinear    space   AFI.      Hence  the   rectangle 

12 


H 

1 

H     B 

/ 
/ 
/ 
/ 
/ 

1 

/ 

G 

y 

E      C 

FlG.    IO. 


1/8  LIMITS    AND   FLUXIONS 

of  variable  base  and  altitude  AEIF  with  the  vertex 
I  moving  along  the  curve  through  B  has  the  fluxion 

In  a  footnote  Rowe  expressed  the  belief  that  this 
mode  of  deriving  the  rule  is  not  open  to  criticism  as 
was  the  method  of  using  increments  vvhich  in  1 734was 
**smartly  attacked  by  the  late  acute  Dr.  Berkeley." 

Rowe  proves  by  a  geometrical  method  similar  to 
the  above  that  the  fluxion  of  a  pyramid  of  fixed 
vertex  and  slant  edges,  whose  variable  base  xy 
moves  parallel  to  itself  and  whose  variable  altitude 
is  z^  is  xyz.  Taking  a  parallelopipedon  as  equal  to 
three  pyramids,  he  finds  the  fluxion  of  xyz  to  be 
xyz-\-xyz-\-xyz.  This  new  way  of  deriving  the 
fluxion  of  xyz  was  copied  by  *'his  friend"  Benjamin 
Martin  in  the  MatJwnatical  Institutions. 

At  the  end  of  the  third  edition  of  Rowe's  Fluxions 
is  a  bibliography  of  English  works  on  this  subject, 
and  he  **particularly  refers  to  the  Works  of  his  two 
celebrated  Friends,  Mr.  Emerson  and  the  late  Mr. 
Simpson." 

Berkeley  Ten  Years  After 
i6i.  Berkeley,  in  his  Siris  ^  of  1744,  expressed 
himself  as  follows  :  "  Concerning  absolute  space, 
that  phantom  of  the  mechanic  and  geometrical 
philosophers  (§  250),  it  may  sufiìce  to  observe  that 
it  is  neither  perceived  by  any  sense,  nor  proved  by 
any  reason,  and  was  accordingly  treated  by  the 
greatest  of  the  ancients  as  a  thing  merely  visionary. 

^  George    Bcrkelcy's    Works.    Edition    by   A.    C.    l'raser,    voi.    ii, 
Oxford,  1871,  p.  468  and  note. 


TEXT-BOOKS,    1736-1741  179 

From  the  notion  of  absolute  space  springs  that  of 
absolute  motion  ..."  He  continues  in  a  foot- 
note  :  "  Our  judgment  in  these  matters  is  not  to  be 
overborne  by  a  presumed  evidence  of  mathematica! 
notions  and  reasonings,  since  it  is  plain  the  mathe- 
maticians  of  this  age  embrace  obscure  notions,  and 
uncertain  opinions,  and  are  puzzled  about  them, 
contradicting  each  other  and  disputing  like  other 
men  :  witness  their  doctrine  of  Fluxions,  about 
which,  within  these  ten  years,  I  bave  seen  published 
about  twenty  tracts  and  dissertations,  whose  authors 
being  utterly  at  variance,  and  inconsistent  with  each 
other,  instruct  by-standers  what  to  think  of  their 
pretensions  to  evidence." 

Remarks 

162.  In  these  publications  no  reference  is  made 
to  the  Jurin-Robins  controversy,  though  Berkeley's 
Analyst  is  frequently  discussed.  Excepting  only  in 
Benjamin  Martin,  the  definition  of  a  fluxion  as  a 
'' differential  "  nowhere  appears.  Therein  we  see 
a  step  in  advance. 

The  influence  of  Newton's  Quadrature  of  Curvcs 
(1704)  is  evident  almost  everywhere.  An  improve- 
ment  in  the  mode  of  deriving  the  fluxion  of  a 
*'  product  "  appears  in  the  anonymous  Explanation  of 
Fluxions  and  in  the  revised  text  by  John  Rowe 
(our  §§  158,   160). 

Noteworthy  is  Thomas  Simpson's  new  definition 
of  fluxions  ;  this  new  definition  plays  an  important 
ròle  during  the  rest  of  the  century. 


i8o  LIMITS    AND   FLUXIONS 

163.  We  quote  Sir  William  Rowan  Hamilton's 
remarks  on  the  lemma  of  the  anonymous  Explana- 
tion  of  Fluxions  (1741)  and  the  derivation  of  the 
fluxion  of  xy^  based  upon  it.  Hamilton  knew  this 
proof  as  it  is  given  in  a  later  edition  of  Simpson's 
fluxions.  Says  Hamilton  :^  *'  I  notice  that  Thomas 
Simpson  treats  fluxions  as  finite  .  .  .  Thomas 
Simpson's  conceptions  appear  to  have  been  very 
clear  and  distinct,  and  I  do  not  venture  to  say  that 
the  geometrical  investigation  which  he  gives  of  the 
fluxion  of  a  rectangle,  avowedly  supplied  to  him  by 
a  young  but  unnamed  friend,  is  insufficient  in  itself, 
but  it  fails  to  convince  me,  perhaps  because  I  was 
not  early  accustomed  to  fluxions.  Certainly  there 
is  no  neglecting  of  ab^  or  .1  j>  ^'^  small  ;  for  in  fact 
that  rectangle  of  the  fluxions  is  not  represented  at 
ali  in  his  Figure  ...  He  conceives  the  varying 
rectangle  xy  to  be  the  suin  of  two  mixtilinear  triangles, 
of  which  the  two  separate  fluxions  are  yx  and  xy. 
This  is  very  ingenious,  but  I  do  not  feel  sure  to 
what  degree  I  could  rely  on  it  and  build  upon  it  any 
superstructure,  if  I  were  now  coming,  for  the  first 
time,  as  a  learner^  to  the  subject.  However,  I 
suppose  that  a  pupil,  if  reasonably  modest  or  even 
prudent,  vvill  take,  for  a  while,  his  teacher's  state- 
ments  upon  trust  ;  reserving  to  himself  to  return 
upon  them,  and  to  examine  closely  their  truth  and 
logie  when  he  shall  have  acquired  some  degree  of 
familiarity  with  the  subject  taught." 

^  Life  of  Sir  Williain  Rowan  Hamillon,  by  R.  P.  Graves,  voi.  iii,  p.  571. 


CHAPTER   VI 

MACLAURIN'S   TREATISE   OF   FLUXIONS,    1742 

164.  Colin  Maclaurin  was  educateci  at  the  Uni- 
versity of  Glasgow,  and  through  the  influence  of 
Newton  was  elected  professor  at  the  University  of 
Edinburgh.  Maclaurin's  hook  on  fluxions  has  been 
considered  the  ablest  and  most  rigorous  text  of 
the  eighteenth  century.  It  was  pronounced  by 
Lagrange  "le  chef  d'oeuvre  de  geometrie  qu'on 
peut  comparer  à  tout  ce  qu'  Archimede  nous  a 
laissé  de  plus  beau  et  de  plus  ingénieux."^ 

In  the  preface  to  his  Trcatise  of  Fluxions^ 
Maclaurin  says  :  "A  Letter  published  in  the  Year 
1734,  under  the  Title  of  the  Analyst,  first  gave 
Occasion  to  the  ensuing  Treatise.  ...  In  the 
mean  Time  the  Defence  of  the  Method  of  Fluxions, 
and  of  the  great  Inventor,  was  not  neglected. 
Besides  an  Answer  to  the  Analyst  that  appcared 
very  early  under  the  Name  of  Philalethes  Canta- 
brigiensis  ...  a  second  by  the  same  Hand  in 
Defence  of  the  first,  a  Discourse  by  Mr.  Robins,  a 

^  Mém.  de  tAcad.  de  Berlin,  1773  ;  quoted  in  the  art.  "  Maclaurin  " 
in  Sidney  Lee's  Dict .  of  National  Biop-aphy. 

^  A  Treatise  of  Fluxions  in  'fwo  Books.  By  Colin  MacLaurin,  A.M., 
Professor  of  Mathematics  in  the  University  of  Edinburgh,  and  Fellow 
of  the  Royal  Society.     Edinburgh,  MDCCXLU. 


i82  LIMITS   AND   FLUXIONS 

Treatise  of  Sir  Isaac  Newton's  with  a  Commentary 
by  Mr.  Colson,  and  several  other  Pieces  were 
published  on  this  Subject.  After  I  saw  that  so 
much  had  been  written  upon  it  to  so  good  Purpose  ; 
I  was  the  rather  induced  to  delay  the  Publication  of 
this  Treatise,  till  I  could  finish  my  Design.  .  .  . 
The  greatest  Part  of  the  first  Book  was  printed  in 
1737  ;  But  it  could  not  bave  been  so  useful  to  the 
Reader  without  the  second.  .  .  .  In  explaining  the 
Notion  of  a  Fluxion,  1  bave  foUowed  Sir  Isaac 
Newton  in  the  first  Book  ...  ;  nor  do  I  think 
that  I  bave  departed  from  bis  Sense  in  the  second 
Book  ;  and  in  both  I  bave  endeavoured  to  avoid 
several  Expressions,  which,  though  convenient, 
might  be  liable  to  Exceptions,  and,  perhaps,  occasion 
Disputes.  I  bave  always  represented  Fluxions  of 
ali  Orders  by  finite  Ouantities,  the  Supposition  of 
an  infinitely  little  Magnitude  being  too  bold  a 
Postulatum  for  such  a  Science  as  Geometry.  But, 
because  the  Method  of  Infinitesimals  is  much  in 
use,  and  is  valued  for  its  Conciseness,  I  thought  it 
was  requisite  to  account  explicitly  for  the  Truth, 
and  perfect  Accuracy,  of  the  Conclusions  that  are 
derived  from  it  .    .   .  " 

165.  In  the  Introduction  to  bis  Fluxions  Maclaurin 
says  :  "...  When  the  certainty  of  any  part  of 
geometry  is  brought  into  question,  the  most  efìfectual 
way  to  set  the  truth  in  a  full  light,  and  to  prevent 
disputes,  is  to  deduce  it  from  axioms  or  first  prin- 
ciples  of  unexceptionable  evidence,  by  demonstra- 
tions  of  the  strictest  kind,  after  the  manner  of  the 


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31.       1 

MACLAURIN'S    TREATISE,    1742      183 

antient  geometricians.  This  is  our  design  in  the 
following  treatise  ;  wherein  we  do  not  propose  to 
alter  Sir  Isaac  Newton's  notion  of  a  fluxion,  but  to 
explain  and  demonstrate  his  method,  by  deducing 
it  at  length  from  a  few  self-evident  truths,  in  that 
strict  manner  :  and,  in  treating  of  it,  to  abstract 
from  ali  principles  and  postulates  that  may  require 
the  imagining  any  other  quantities  but  sudi  as  may 
be  easily  conceived  to  have  a  real  existence.  We 
shall  not  consider  any  part  of  space  or  time  as 
indivisible,  or  infinitely  little  ;  but  we  shall  consider 
a  point  as  a  term  or  limit  of  a  line,  and  a  moment 
as  a  term  or  limit  of  time  ...  [p.  41].  If  we  are 
able  to  join  infinity  to  any  supposed  idea  of  a  deter- 
minate quantity,  and  to  reason  concerning  magni- 
tude  actually  infinite,  it  is  not  surely  with  that 
perspicuity  that  is  required  in  geometry.  In  the 
same  manner,  no  magnitude  can  be  conceived  so 
small,  but  a  less  than  it  may  be  supposed  ;  but 
we  are  not  therefore  able  to  conceive  a  quantity 
infinitely  small  ..." 

166.  In  the  posthumous  work,  An  Account  of  Sir 
Isaac  NewtoìCs  Philosophical  Discoveries^  by  Colin 
Maclaurin,  2nd  ed.,  London,  1750,  there  is  printed 
a  life  of  Maclaurin,  from  which  we  glean  the  follow- 
ing (pp.  viii,  ix,  and  xviii)  relating  to  Berkeley 's 
attack  in  the  Analyst  : 

"  Mr.  Maclaurin  found  it  necessary  to  vindicate 
his  favourite  study,  and  repel  an  accusation  in  which 
he  was  most  unjustly  included.  He  began  an 
answer  to  the  bishop's  hook  ;  but  as  he  proceeded, 


i84  LIMITS    AND    FLUXTONS 

so  many  discoveries,  so  many  new  theorìes  and 
problems  occurred  to  him,  that,  instead  of  a  vindi- 
catory  pamphlet,  his  work  carne  out  a  complete 
system  of  fluxions,  with  their  application  to  the 
most  considerable  problems  in  geometry  and  naturai 
philosophy.  This  work  was  published  at  Edinburgh 
in  1742.  .  .  .  Piis  demonstrations  had  been,  several 
years  before,  communicated  to  Dr.  Berkeley,  and 
Mr.  Maclaurin  had  treated  him  with  the  greatest 
personal  respect  and  civility  :  notwithstanding 
which,  in  his  pamphlet  on  tar- water,  ^  he  renews 
the  charge,  as  if  nothing  had  beeh  done  ;  for  this  ex- 
cellent  reason,  that  difìerent  persons  had  conceived 
and  expressed  the  same  thing  in  different  ways.  .  .  . 
Mr.  Maclaurin  found  it  necessary,  in  demonstrating 
the  principles  of  fluxions,  to  reject  altogether  those 
exceptionable  terms  {infinite  and  infinitesimal\  and 
to  suppose  no  other  than  finite  determinale  quan- 
tities,  such  as  Euclid  treats  of  in  his  geometry." 

167.  In  Chapter  1,  p.  57,  Maclaurin  defines  a 
fluxion  :  "  The  velocity  with  which  a  quantity  flows, 
at  any  term  of  the  time  while  it  is  supposed  to  be 
generated,  is  called  its  Fluxion  which  is  therefore 
always  measured  by  the  increment  or  decrement 
that  would  be  generated  in  a  given  time  by  this 
motion,  if  it  was  continued  uniformly  from  that 
term  without  any  acceleration  or  retardation  :  or 
it  may  be  measured  by  the  quantity  that  is  gener- 
ated in  a  given  time  by  an  uniform  motion  which 
is  equal  to  the  generating  motion  at  that  term." 

1  In  the  second  cdition  Berkeley  gave  the  article  the  name  oi  Siiis, 


MACLAURIN'S    TREATISE,    1742      185 

The  term  velocity  had  been  under  dispute,  par- 
ticularly  in  the  controversy  between  Berkeley  and 
Walton.  Maclaurin  evidently  perceived  the  diffi- 
culty  in  arguing  that  variable  velocity  is  a  physical 
fact  ;  he  says  (p.  55),  "the  velocity  of  a  variable 
motion  at  any  given  term  of  time  is  not  to  be 
measured  by  the  space  that  is  actually  described 
after  that  term  in  a  given  time,  but  by  the  space 
that  would  have  been  described  if  the  motion  had 
continued  uniformly  from  that  term.  If  the  action 
of  a  variable  power,  or  the  velocity  of  a  variable 
motion,  may  not  be  measured  in  this  manner,  they 
must  not  be  susceptible  of  any  mensuration  at  ali  "  — 
an  argument  not  lii<ely  to  convince  or  silence  hostile 
critics.  He  quotes  Barrow's  definition  of  velocity  — 
*'the  power  by  which  a  certain  space  may  be 
described  in  a  certain  time."  In  discussing  * 'power" 
Maclaurin  brings  in  the  consideration  of  *  '  cause  " 
and  ''efìfect"  in  a  way  that  sounds  odd  in  a  work 
laying  the  foundations  to  the  abstract  doctrine  of 
fluxions.  Maclaurin  uses  the  word  **  limit,"  without 
giving  it  a  formai  definition.  Theorem  XII  reads  : 
'*The  velocity  of  a  motion  that  is  accelerated  or 
retarded  perpetually,  is,  at  any  term  of  the  time, 
to  the  velocity  of  an  uniform  motion,  in  a  ratio  that 
is  always  a  limit  between  the  ratio  of  the  spaces 
described  by  these  motions  in  any  equal  times 
before  that  term,  and  the  ratio  of  the  spaces  de- 
scribed by  them  in  any  equal  times  after  it.  " 

168.  In  the  Pliilosophical  Trans acttons^  voi.  xlii, 
for  the   years    1742-43,   London,    1744,    Maclaurin 


i86  LIMITS    AND   FLUXIONS 

gives  an  account  of  his  Treatise  of  Fluxions.  On 
p.  330  of  these  Transactions  it  is  pointed  out  that 
"the  Theory  of  Motion  is  rendered  applicable  to 
this  Doctrine  with  the  greatest  Evidence,  without 
supposing  Quantities  infinitely  little  or  having 
recourse  to  prime  or  ultimate  Ratios."  Again 
(P-  336):  *'There  is,  however,  no  Necessity  for 
considering  Magnitude  as  made  up  of  an  infinite 
Number  of  small  Parts  ;  it  is  sufficient,  that  no 
Quantity  can  be  supposed  to  be  so  small,  but  it 
may  be  conceived  to  be  diminished  further  ;  and  it 
is  obvious,  that  we  are  not  to  estimate  the  Number 
of  Parts  that  may  be  conceived  in  a  given  Magni- 
tude, by  those  which  in  particular  determinate 
Circumstances  may  be  actually  perceived  in  it  by 
Sense  ;  since  a  greater  Number  of  Parts  become 
visible  in  it  by  varying  the  Circumstances  in  which 
it  is  perceived."  Of  importance  is  the  lollowing 
(p.  336):  "  We  shall  therefore  observe  only,  that 
after  giving  some  plain  and  obvious  Instances, 
wherein  a  Quantity  is  always  increasing,  and  yet 
never  amounts  to  a  certain  finite  Magnitude  (as, 
while  the  Tangent  increases  the  Are  increases  but 
never  amounts  to  a  Ouadrant)."  That  a  variable 
need  not  reach  its  limit  is  also  emphasised  in  other 
passages,  as  for  instance  (pp.  337,  338):  '*  In  like 
manner  a  curvilineal  Area  .  .  .  may  increase,  while 
the  base  is  produced,  and  approach  continually  to  a 
certain  finite  Space,  but  never  amount  to  it.  .  .  . 
A  Spirai  may  in  like  manner  approach  to  a  Point 
continually,  and  yet  in  any  Number  of  Revolutions 


MACLAURIN'S    TREATISE,    1742      187 

never  arrive  at  it.  .  .  .  The  Author  insists  on  these 
Subjects,the  rather  that  they  are  commonly  described 
in  very  mysterious  Terms,  and  have  the  most  fertile 
of  Paradoxes  of  any  Parts  of  the  higher  Geometry.'* 

The  ideal  of  mathematica!  rigour,  as  entertained 
by  eighteenth-century  writers,  was  reached  by  the 
Greek  geometricians,  Euclid  and  Archimedes.  To 
derive  the  rules  of  fluxions  by  the  rigorous  methods 
of  the  ancients  was  the  ambition  of  Maclaurin. 
Barring  some  obvious  slips  that  are  easily  remedied, 
Maclaurin  certainly  reached  the  ideal  he  had  set. 
Nor  is  this  so  very  strange.  Fluxions  involve 
questions  concerning  limits  ;  the  ancients  overcame 
the  difficulties  of  such  questions  by  their  method  of 
exhaustion.  It  was  a  rigorous  method,  but  dread- 
fully  tedious.  Maclaurin  secured  his  aim  at  a 
tremendous  sacrifice.  His  work  on  fluxions  consists 
of  763  good-sized  pages  ;  the  first  590  pages  do  not 
contain  the  notation  of  fluxions  at  ali  ;  they  deal 
with  the  derivation  of  the  fluxions  of  different 
geometrie  figures,  of  logarithms,  of  trigonometrie 
functions,  also  with  the  discussions  of  maxima  and 
minima,  asymptotes,  curvature,  and  mechanics,  in 
a  manner  that  the  ancients  might  have  followed, 
and  with  the  verbosity  of  which  the  ancients  are 
guilty.  The  consequence  was  that  the  work  was 
not  attractive  reading. 

Maclaurin  was  fuUy  aware  of  the  value  of  a  good 
notation  and  case  of  operation,  for  he  says  of  the 
doctrine  (p.  575):  ''The  improvements  that  have 
been  made  by  it,   either  in  geometry  or   in   philo- 


i88  LIMITS   AND   FLUXIONS 

sophy,  are  in  a  great  measure  owing  to  the  facility, 
conciseness  and  great  extent  of  the  method  of  com- 
putation, or  algebraic  part.  It  is  for  the  sake  of 
these  advantages  that  so  many  symbols  are  employed 
in  algebra."  But  to  Maclaurin  it  seemed  "  worth 
while  to  demonstrate  the  chief  propositions  of  this 
method  in  as  clear  and  compleat  a  manner  as 
possible,  if  by  this  means  we  can  preserve  this 
science  from  disputes  "  (p.  102).  We  shall  see  that 
Maclaurin's  book  did  not  stop  disputes.  Had  the 
book  been  read  more,  it  might  have  been  more 
effective  in  this  respect.  Our  studies  have  led  us 
to  the  conclusion  that  Maclaurin  was  not  widely 
read.  A  second  edition  of  his  F/uxions  did  not 
appear  until  1801.  His  work  was  praised  highly, 
but  seldom  used  and  digested.  We  might  say  of 
Maclaurin  vvhat  has  been  said  of  the  German  poet 
Klopstock  : — 

"Wer  wird  nicht  einen  Klopstock  loben  ? 
Uoch  wird  ihn  jeder  lesen  ? — Nein. 
Wir  wollen  weniger  erhoben, 
Und  fleissiger  gelesen  seyn." 

Remarks  ^ 

169.    To  what  extent,  if  any,  Maclaurin  may  have 
been  influenced  by  Robins  in  the  mode  of  trcating 

^  In  1745  there  appeared  an  anonymous  publication  on  fluxions 
which  we  have  not  had  the  opportunity  lo  examine  ;  it  was  entitled, 
The  Harmony  of  the  Ancient  and  Modem  Geometry  asserted.  In 
A.  C.  Fraser's  edition  of  Berkeley's  Works,  voi.  iii,  Oxford,  187 1, 
p.  301,  it  is  referred  to  as  follows  :  "This  last  and  forgotten  tract 
consists  of  papers  given  in  to  the  Royal  Society  in  1742,  and  treats 
fluxions  as  a  particular  branch  of  an  alleged  more  general  reasoning, 
called  the  doctrine  of  maximinority  and  minimajority," 


MACLAURIN'S    TREATISE,   1742      189 

fluxions  it  is  difficult  to  say.  Later  we  shall  see 
what  James  Wilson  states  on  this  point.  Certain 
it  is  that  Maclaurin's  views  agree  much  more  closely 
with  those  of  Robins  than  those  of  Jurin.  Maclaurin 
stood  for  the  narrower  view  of  limits — limits  none 
of  which  are  reached  by  the  variable.  However, 
the  conception  of  Hmits  does  not  receive  as  great  a 
degree  of  emphasis  with  Maclaurin  as  it  does  in  the 
Discoursc  of  Robins. 

Of  Maclaurin's  Fluxions^  Professor  Kelland  has 
remarked  :  "The  Analyst  did  good  service  to 
science,  if  in  no  other  way,  at  least  by  giving 
occasion  to  this  last  work.  The  principles  of  the 
method  had  been  previously  exhibited  in  a  concise 
and  obscure  manner  ;  Maclaurin  developed  them 
after  the  manner  of  ancient  geometers." 

In  1749,  Maclaurin's  Treatisc  of  Fluxions  was 
translated  into  French  by  Esprit  Pezenas,  director 
of  the  observatory  at  Avignon. 

As  we  look  back,  we  see  that  the  eight  years  ìm- 
mediately  foUowing  Berkeley's  A^talyst  were  eight 
great  years,  during  which  Jurin,  and  especially 
Robins  and  Maclaurin,  made  wonderful  progress  in 
the  banishment  of  infinitely  small  quantities  and  the 
development  of  the  concept  of  a  limit.  Both  before 
and  after  that  eight-year  period  there  were  published 
books  in  Great  Britain  containing  a  mixture  of  Con- 
tinental and  British  conceptions  of  the  new  calculus, 
a  superposition  of  British  symbols  and  phraseology 
upon  the  older  Continental  concepts. 


CHAPTER  VII 

TEXT-BOOKS   OF  THE    MIDDLE   OF 
THE   CENTURY 

John  Stewai't,   1745 

170.  John  Stewart,  professor  of  mathematics  at 
Marischal  College  in  Aberdeen,  is  known  as  the 
translator  into  English,  with  commentaries,  of 
Newton's  Quadì-ature  of  CtU'ves  and  Analysis  by 
Equations  of  an  Infinite  Nuniber  of  Terms.^ 

The  translator  spares  no  pains  in  the  endeavour 
to  remove  any  obscurities  which  the  ordinary  reader 
might  encounter  therein.  Newton's  Quadrature  of 
Curves  takes  up  33  pages  in  John  Stewart's  volume; 
Stewart's  explanations  thereof  fili  287  pages.  Re- 
ferring  to  the  controversy  between  Berkeley  and 
Jurin,  Stewart  says  that  ''because  the  Doctrine  of 
prime  and  ultimate  Ratios  has  been  so  much  con- 
troverted  of  late,  I  shall  bere  enquire  whether 
we  bave  any  distinct  Idea  thereof."  He  quotes 
Newton's  Lemma  i  in  Book  I,  Section  i  of  the 
Principia^  also  the  proof  of  it,  and  then  argues  that 
the  limit   is  reached,    for   **a   Difference    less   than 

'  Sir  Isaac  Newton^ s  Two  Treaiises  of  the  Quadrature  of  Curves,  and 
Analysis  by  Equations  of  an  Infinite  Nutnber  of  Terms,  explained. 
By  John  Stewart,     London,  1745. 

190 


TEXT-BOOKS  OF MIDDLE  OFCENTUR  F  191 

any  Thing  assignable,  is  the  same  Thing  as  no 
Difference  at  ali  :  for  repeat  it  as  often  as  you 
please,  it  can  never  be  equal  to  any  finite  Ouantity  : 
and  therefore  can  bear  no  Ratio  to  it,  by  Def.  4, 
Bk.  5  [of  Euclid's]  Elements"  (p.  37).  Stewart 
gives  definitions  of  ultimate  ratio  of  quantities  and 
of  evanescent  quantities,  also  definitions  of  prime 
ratio  of  quantities  and  of  nascent  quantities.  The 
foUowing  is  a  specimen:  ''The  ultimate  Ratio  of 
evanescent  Quantities  is  the  Limit  to  which  the 
Ratio  of  variable  Quantities  diminishing  without 
Bound,  continually  approaches,  to  come  nearer  to 
it  than  by  any  given  Difference  ;  but  which  never 
goes  beyond  ;  yet  no  sooner  attains  to,  than  the 
Quantities  being  diminished  infinitely,  vanish."  The 
following  additional  statement  follows  closely  the 
language  of  Newton  (p.  39):  '' If  any  one  should 
object  that  there  can  be  no  ultimate  Ratio  of 
continually  diminishing  and  at  last  evanescent 
Quantities  :  because  before  they  vanish  it  is  not  the 
last  ;  and  after  they  vanish,  they  have  no  Ratio. 
The  Answer  is,  that  the  ultimate  Ratio  is  neither 
the  Ratio  of  them  before  they  vanish  ;  nor  after 
they  vanish  ;  but  the  Ratio  wherewith  they  vanish, 
or  the  Limit  to  which  their  varying  Ratio  no  sooner 
arrives,  than  they  vanish  ;  .  .  .  that  Ratio  they 
have  that  very  Instant  they  vanish.  .  .  .  It  signi- 
fies  nothing  to  say  ultimate  Quantities  cannot  be 
assigned,  in  regard  Quantity  is  divisible  without 
End  :  for  it  is  not  the  Quantities  themselves  that 
are    hereby     determined,     but    only    their     Ratio  : 


192  LIMITS    AND   FLUXIONS 

which  is  capable  of  being  determined."  This 
matter,  says  Stewart,  has  been  so  clearly  explained 
by  Newton,  *'that  the  great  Dust  which  has  been 
raised  of  late  about  the  Whole  of  this  Doctrine, 
must  be  owing  to  Weakness,  or  some  worse 
Principle  "  (p.  40). 

William  Emerson,   1743  (?),   1757,  1768 

171.  William  Emerson  was  a  self-taught  mathe- 
matician  ;  he  wrote  many  mathematical  texts  which 
indicate  a  good  grasp  of  existing  knowledge,  but  not 
great  originality.  His  Doctrine  of  Fluxions  appeared 
at  London  in  1743  (?).  We  bave  before  us  the  third 
edition,   1768.      From  it  we  quote  as  follows  : 

'*The  Velocity  of  the  Increase  of  any  generated 
Ouantity,  or  the  Degree  of  Quickness  (or  Slowness) 
wherewith  the  new  Parts  of  it  continually  arise,  is 
called  its  Fluxion.  " 

"The  indefinitely  small  Portions  of  the  Pluent 
which  are  generated  in  any  indefinitely  small  Por- 
tions of  Time  are  called  Moments  or  Increments." 

**.  .  .  The  Moments  and  Fluxions  ought  not  to 
be  confounded  together,  since  the  Moments  (being 
generated  by  Fluxions)  are  as  difìferent  from  the 
Fluxions,  as  any  Effect  is  different  from  its  Cause." 

The  following  is  given  as  an  axiom  : 

"  Ouantities,  which  in  any  finite  Time  continually 
converge  to  Equality,  and  before  the  P^nd  of  that 
Time,  approach  nearer  to  one  another  than  by  any 
given  Difference,  do  at  last  become  equal." 

"  If  any  should  think   this  not  clear  enough   to 


TEXT-BOOKS  OF MIDDLEOF CENTUR  Y  193 

pass  for  an  Axiom,  he  may  consider  it  thus  ;  let 
D  be  their  ultimate  Difference,  therefore  they 
cannot  approach  nearer  to  equality,  than  by  that 
given  Difference  D,  contrary  to  the  Hypothesis  ; 
which  Supposition  is  absurd  in  ali  Cases  except 
when  D  is  nothing." 

To  find  the  fluxion  of  òx"y"\  he  lets  ox,  oy  be 
moments,  expands  the  powers  of  x-\-ox  andjz  +  ^j, 
and  finds  the  increments.  Then  he  divides  **by 
the  indefinite  Quantity  0."  "  But  since  the 
(Velocity  or)  Fluxion  is  required  wherewith  that 
Moment  first  arises,  in  this  Case  the  Moments  ox 
and  oj  will  also  be  just  arising  and  therefore 
nothing,  and  consequently  0  will  be  nothing,  and 
therefore  ali  the  Terms  wherein  it  is  found  will  be 
nothing."  The  final  result  then  foUows.  In  his 
Preface  Emerson  claims  that  **  Velocity  must  be 
looked  upon  as  the  proper  efìficient  Cause  of  the 
Space  described  ;  and  the  Space  described  the 
adequate  Effect  of  that  Cause."  .  .  .  **No  incre- 
ment  can  be  taken  so  small,  but  it  is  stili  further 
divisible  ad  infinitum  ;  and  since  the  Velocity  is 
by  Supposition  continually  variable,  it  is  plain, 
there  can  be  no  two  Points  of  the  Increment  in 
both  of  which  the  Velocity  is  accurately  the  same. 
It  is  therefore  most  manifest,  that  the  Velocity  bere 
enquired  after  is  peculiar  to  one  only  indivisible 
Point  ;  .  .  .  that  the  Velocity  in  any  given  Point 
of  the  Line  described  .  .  .  has  a  certain,  fixed. 
determinate  Value.  .  .  .  Here  a  metaphysical 
Disputant  may  demand,  how  it  comes  to  pass,  that 

13 


194  LIMITS    AND   FLUXIONS 

aiiy  Velocity  which  continues  for  no  Time  at  ali,  can 
possibly  describe  any  Space  at  ali  ;  or  whether  its 
Effect  be  absolutely  nothing,  or  an  infinitely  small 
Ouantity,  or  what  it  is.  Here  then  it  is,  that  our 
Reason  is  at  a  Stand,  and  the  human  Faculties  are 
quite  confounded,  lost,  and  bewildered.  .  .  .  Now 
whether  such  subtile  Questions  will  be  ever  de- 
termined,  or  not,  yet  there  is  one  Refuge  for  us, 
viz.  that  it  is  nothing  at  ali  to  our  Purpose  what 
they  are  :  .  .  .  The  Method  of  Fluxions  has 
no  Dependence  on  these  mysterious  Disquìsitions. 
What  I  apprehend  the  Method  of  Fluxions  to  be 
concerned  in,  is  .  .  .  what  a  .  .  .  variable  Velocity 
can  produce  in  the  whole.  And  here  1  think  no 
Reason  can  be  assigned,  why  a  variable  Cause 
•should  not  produce  a  variable  Effect,  .  .  .  though 
we  have  no  Ideas  at  ali  of  the  perpetually  arising 
Increments,  or  their  Magnitude  in  their  nascent  or 
evanescent  State,  that  have  so  much,  and  to  so  little 
Purpose,  confounded  and  puzzled  the  mathematical 
World." 

Thomas  Simpsoji,    1750 

172.  Simpson's  Ti-eatise  of  Fluxions  of  1737  has 
already  been  noticed  (our  §  156).  His  text  of 
1750,  The  Doctrine  and  Application  of  Fluxions, 
London,  is  new,  not  only  in  the  title,  but  to  some 
extent  also  in  the  mode  of  exposition.  He  says  in 
his  Preface  (1750)  that  he  has  used  a  tract  entitled 
An  Explanation  of  Fluxions  in  a  Short  Essay  on  the 
Theory,  printed  by  W.  Innys  and  written  by  one  of 


TEXT-BOOKS  OF MIDDLEOF CENTUR  V  195 

his  friends  who  was  too  modest  to  put  his  name  to 
it.  (See  our  §§  158,  160,  163.)  Simpson  used  his 
friend's  manner  of  determining  the  fluxion  of  a  rect- 
angle  and  of  illustrating  fluxions  of  higher  order. 
Simpson  defines  a  fluxion  as  foUows  : 

''  The  Magnitude  by  which  any  Flowing  Ouantity 
would  be  uniformly  increased,  in  a  given  Portion  of 
Time,  with  the  generating  Celerity  at  any  proposed 
Position,  or  Instant  (was  it  from  thence  to  continue 
invariable),  is  the  Fluxion  of  the  said  Ouantity  at 
that  Position,  or  Instant." 

The  derivation  of  the  fluxion  of  xf  is  explained 
after  the  manner  adopted  by  John  Rowe,  both 
authors  being  indebted  for  it  to  the  author  of  An 
Explanation  of  Fluxions  in  a  Short  Essay  on  the 
Theory.  The  same  definitions  and  explanations  of 
the  fundamentals  are  given  by  Thomas  Simpson 
in  the  last  part  of  his  Select  Exerciscs  for  Young 
Proficients  in  the  M athernaticks ,  1752.  In  the 
preface  to  his  Fluxions  of  1750,  Simpson  touches 
some  points  of  philosophic  interest.      He  says  : 

'*  By  taking  Fluxions  as  ineer  Velocities^  the 
Imagination  is  confin'd,  as  it  were,  to  a  Point,  and 
without  proper  Care  insensibly  involv'd  in  meta- 
physical  Difificulties  :  But  according  tò  our  Method 
of  conceiving  and  explaining  the  Matter,  less 
Caution  in  the  Learner  is  necessary,  and  the  higher 
Orders  of  Fluxions  are  render'd  much  more  easy 
and  intelligible — Besides,  tho'  Sir  Isaac  Newton 
defines  Fluxions  to  be  tìic  Velocities  of  Moiions, 
yet    he    hath     Recourse     to     the     Increments,     or 


196  LIMITS    AND   FLUXIONS 

Moments,  generateci  in  equal  Farticles  of  Time,  in 
order  to  determine  those  Velocities  ;  which  he  after- 
wards  teaches  us  to  expound  by  finite  Magnitudes 
of  other  Kinds  :  Without  which  (as  is  already  hinted 
above)  we  could  have  but  very  obscure  Ideas  of 
higher  Orders  of  Fluxions  :  For  if  Motion  in  (or  at) 
a  Foint  be  so  difficult  to  conceive,  that  Some  have, 
even,  gone  so  far  as  to  dispute  the  very  Existence 
of  Motion,  how  much  more  perplexing  must  it  be 
to  form  a  Conception,  not  only,  of  the  Velocity  of  a 
Motion,  but  also  infinite  Changes  and  Affections  of 
It,  in  one  and  the  same  Point,  where  ali  the  Orders 
of  Fluxions  are  to  be  considered. 

"  Seeing  the  Notion  of  a  Fluxion,  according  to 
our  Manner  of  defining  It,  supposes  an  Uniform 
Motion,  it  may,  perhaps,  seem  a  Matter  of  Diffi- 
culty,  at  first  View,  how  the  Fluxions  of  Quantities, 
generated  by  Means  of  accelerated  and  retarded 
Motions,  can  be  rightly  assigned  ;  since  not  any, 
the  least,  Time  can  be  taken  during  which  the 
generating  Celerity  continues  the  same  :  Here, 
indeed,  we  cannot  express  the  Fluxion  by  any 
Increment  or  Space,  actually  generated  in  a  given 
Time  (as  in  uniform  Motion).  But,  then,  we  can 
casily  determine,  what  the  contemporary  Increment, 
or  generated  Space  would  be^  if  the  Acceleration,  or 
Retardation,  was  to  cease  at  the  proposed  Position 
in  which  the  Pluxion  is  to  be  found  :  Whence  the 
true  Fluxion,  itself,  will  be  obtained,  without  the 
Assistancc  of  infinitely  small  Quantities,  or  any 
metaphysical  Considerations.  " 


TEXT-BOOKS  OF MIDDLE  OFCENTUR  Y  197 

Nicholas  S aunder soìi,    1756 

173.  At  the  age  of  twelve  months  Saunderson 
lost  his  eyesight  by  small-pox  ;  nevertheless,  he 
rose  to  prominence.  He  studied  at  Christ's  College, 
Cambridge,  and  in  171 1  succeeded  VVhiston  as 
Lucasian  professor  of  mathematics  at  Cambridge. 
His  Fluxions  ^  is  a  posthumous  vvork. 

VVe  read  (p.  i)  :  *' Let  AB  represent  any 
Moment  of  Time,  whether  finite  or  infinitely  small 
it  matters  not,  terminated  by  the  two  Instants 
A  and  B.  Let  x  be  the  Value  of  any  flowing  or 
growing  Quantity  at  any  Instant  A,  whose  Velocity 
at  that  Instant  is  such,  that  if  it  was  to  flow  during 
the  whole  Moment  AB  with  this  Velocity,  it  would 
gain  a  certain  Increment  represented  by  x  ;  then  is 
this  Quantity  x  called  the  Fluxion  of  x  at  the 
Instant  A,  when  the  Value  of  the  flowing  Quantity 
was  X.''  In  the  scholium  which  follows,  it  is 
explained  that  if  the  velocity  is  variable,  then  the 
increment  of  the  velocity  "  gained  in  the  time  AB 
will  not  be  the  same  with  its  Fluxion  above  defined, 
.  .  .  but  if  the  Time  AB  be  infinitely  small,  then 
though  the  Velocity  of  x  at  the  Instant  B  be 
not  the  same,  mathematically  speaking,  with  the 
Velocity  at  the  Instant  A  ;  yet  the  Difference  being 
infinitely  small  in  Respect  of  the  whole  Velocity,  it 
may  safely  be  neglected,  where  the  finite  Ratios  of 
Fluxions  are  only  considered  ;  and  so  this  Increment 

^  The  Method  of  Fluxions  Applied  to  a  select  Number  of  Useful 
Problems.  .  .  .  By  Nicholas  Saunderson,  Late  Professor  of  Mathematics 
in  the  University  of  Cambridge.     London,  MDCCLVL 


198  LIMITS    AND   FLUXIONS 

and  the  Fluxion  above  defined  may  be  taken  for 
one  another,  i.e.  the  Ouantity  x  for  so  small  a 
Time,  may  be  looked  upon  as  flowìng  uniformly  " 
(p.  2).  Later  we  read  (p.  4)  that  if  the  times  are 
infinitely  small,  the  quantity  z>i- will  be  "infinitely 
less  "  than  vx  or  xi).  Here  the  fluxions  x,  v,i\  are 
looked  upon  as  mfinitely  small. 

In  the  account  of  the  life  of  Nicholas  Saunderson, 
printed  in  the  first  volume  of  his  Elements  of  Algebra ^ 
Cambridge,  17^0,  p.  xv,  we  read:  "  Our  Professor 
would  not  be  induced  by  the  Desires  and  Expecta- 
tions  of  any,  to  engagé  in  the  war  that  was  lately 
waged  among  Mathematicians,  vvith  no  small  Degree 
of  Heat,  concerning  the  Algorithm  or  Principles  of 
Fluxiotis.  Yet  he  wanted  not  the  greatest  Respect 
for  the  Memory  of  Sir  Isaac  Newton,  and  thought 
the  whole  Doctrine  entirely  defensible  by  the  strictest 
Rules  of  geometry.  He  owned  indeed  that  the 
great  Inventor,  never  expecting  to  bave  it  canvassed 
with  so  much  trifling  Subtility  and  Cavil,  had  not 
thought  it  necessary  to  be  guarded  every  where  by 
Expressions  so  cautious  as  he  might  bave  otherwise 
used." 

folm  Rowning,  1756 

1 74.  A  graduate  of  Magdalene  College,  Cambridge, 
and  a  Fellow  there,Rowning  interested  himself  chiefly 
in  naturai  philosophy,  but  wrote  also  A  Prelinmiary 
Discoiirse^  on  fluxions,  with  the  intention  of  writing 

^  A  Pieliminary  Discotirse  to  an  ititended  Treatise  on  the  Fluxiouary 
Method.      Hy  John  Rowning,  M.A.     London,  1756. 


TEXT-BOOKS  OF MIDDLE  OF  CE N TUR  Y 199 

a  full  treatise.  But  the  treatise  in  question  never 
appeared.  After  a  popular  exposition  of  the  ideas 
of  fluxion  and  fluent,  and  of  Leibniz's  infinitely  little 
quantities  and  their  summation,  showing  how  these 
methodsyield  importantresults  in  naturai  philosophy, 
he  refers  to  Berkeley's  attacks  and  the  defence  made 
by  Philalethes  Cantabrigiensis,  Walton,  and  Robins, 
also  Maclaurin,  who  '  *  declined  entering  the  Combat," 
but  endeavoured  to  treat  the  subject  *'in  a  Manner 
less  exceptionable."  *'But  no  Body,  that  I  know 
of,"  continues  Rowning,  '*has  explained  it  in  so 
easy  and  familiar  a  VVay  as  I  apprehend  the  Subject 
capable  of."  Moreover,  Jurin  and  Walton  "  carry 
things  ...  no  farther  than  Sir  Isaac  had  done 
before.  They  leave  them,  as  to  the  Objections 
made  by  the  Analyst,  exactly  as  they  found  them." 
The  diffìculties  do  not  He  in  the  idea  of  a  first 
fluxion — a  velocity.  *'  In  this  there  is  Nothing 
either  infinitely  great  or  infinitely  little  :  Nothing 
obscure. "  As  to  higher  fluxions,  "these  Things 
indeed  elude  our  Senses  ;  but  they  do  not  surpass 
the  Understanding  "  (p.  85).  Berkeley's  objection 
to  'Mnfinitely  small  Quantities"  is  not  fatai, 
"  because  finite  Measures  might  bave  been  made 
use  of."  His  other  objection,  that  "  such  Quanti- 
ties are  in  some  Cases  retained  and  made  use  of  for 
a  while,  and  afterwards,  to  use  his  own  Expression, 
like  Scaffolds  to  a  Building,  are  rejected  as  of  no 
Significancy,"  may  be  met  by  the  proof  that  those 
quantities  "are  always  such  as  ought  by  no  means 
to    be    retained."       In    further    explanation    of    his 


200  LIMITS   AND   FLUXIONS 

position  Rowning  says  (p.  88),  '*that  the  Velocity 
of  any  Body  is  the  same  at  any  one  Point,  or  at  any 
one  Time,  whether  the  Body  moves  with  an  uniform, 
accelerated,  or  with  a  retarded  Motion  at  that  Point 
or  Time."  This  is  elucidated  by  reference  to  geo- 
metrie figures,  and  amounts,  in  the  main,  to  the 
explanation  given  by  Rowe  in  finding  the  fluxion 
of  xy.  One  objection  to  such  explanations,  which 
had  been  raised  by  Berkeley,  was  that  one  could  not 
speak  of  the  velocity  a  body  had  at  a  point  of  space. 
That  such  a  phraseology  is  admissible  is  tacitly 
assumed  by  Rowning.  What  the  latter  emphasises 
is  that  no  use  is  made  of  the  concept  of  the 
*  '  infinitely  little.  "  As  to  Berkeley's  second  objec- 
tion, that  the  supposition  which  is  made  at  the 
beginning  of  the  process  is  later  displaced  by  its 
contrary,  as  when  the  symbol  o  is  at  first  made  an 
actual  increment  and  later  in  the  same  process  taken 
as  no  increment,  Rowning  argues  that  terms  involv- 
ing  factors  oo^  ooo,  etc. ,  "do  arise  in  consequence 
of  the  Acceleration  wherewith  the  Power  of  ;ir  flows, 
when  X  itself  flows  uniformly  ;  and  consequently 
that  they  arise  from  the  second  and  higher  Fluxions 
of  that  Power  ;  and  that,  therefore,  when  the  first 
Fluxion  of  that  power  is  only  inquired  after  .  .  . 
they  are  to  be  left  out  and  rejected,  as  appertain- 
ing  to  another  Account."  It  can  hardly  be  claimed 
that  Rowning  made  a  contribution  to  the  theory 
of  fluxions.  However,  he  has  a  pleasant  way  of 
expressing  himself.  His  book  was  favourably  re- 
viewed  in  the  Monthly  Review  (voi.  xiv,  p.  2 86). 


TEXr-BOOKS  OF MIDDLE  OFCENTUR  Y  201 

Israel  Lyons,   1758 

175.  Lyons  was  a  mathematician  and  botanist. 
His  Trcatise  of  Fluxions,  London,  1758,  is  dedi- 
cated  to  Robert  Smith,  Master  of  Trinity  College, 
Cambridge,  "  being  the  first  Essay  of  a  young  and 
unpractised  Writer  "  which  '  '  ovved  its  first  rude 
Beginning  to  the  early  Encouragement  "  receìved 
from  the  Master,  as  the  author  modestly  states. 
His  treatment  is  geometrie.  He  says  :  "I  reject 
no  Ouantities  as  infinitely  smaller  than  the  rest, 
nor  suppose  difìferent  Orders  of  Infinitesimals  and 
infinitely  great  Ouantities.  But  consider  the  Ratio 
of  the  Fluxions  as  the  same  as  that  of  the  con- 
temporaneous  Increments,  and  take  Part  of  the 
Increment  before  and  Part  after  the  Fluent  is 
arrived  at  the  Term,  where  we  want  the  Fluxion, 
since  it  is  not  the  Increment  after,  or  the  Increment 
before  that  we  want,  but  at  the  very  instant,  which 
can  no  otherwise  be  found  but  by  considering  Part 
of  the  Increment  before  and  Part  after"  (Preface). 
Fluxions  are  defined  as  velocities.  '*The  moments 
of  quantities  are  the  indefinitely  small  parts,  by  the 
addition  or  subtraction  of  which,  in  equal  particles 
of  time,they  are  continually  increased  or  diminished.  " 
The  author  proves  the  proposition  :  **The  indefi- 
nitely small  spaces  described  in  equal  indefinitely 
small  times  are  as  the  velocities,"  since,  "  when 
the  time  is  diminished  ad  infinitum,  the  difference 
of  the  velocities  at  the  beginning  and  ending  of 
that  time  will  vanish."      If  two  flowing  quantities 


202  LIMITS   AND  FLUXIONS 

X  and  y  are  to  each  other  in  a  given  ratio,  then  in 
xy  =  z  it  is  argued  that  2y  =  ìucr.  of  ^-^incr.  of 
x  =  è-^x;  hence  é=2yx.      Whenj'  =  x,  this  becomes 


è=2x±;  one  has  also,  fluxion  x-\-y  =2x-\-vxx-\-y 
=  2  X  xx-\-yx-\-xy  -\-yy. 

From  this  is  derived  the  fluxion  of  any  rectangle 

2 

xy,  thus  :  The  fluxion  of  xy,  or  x'^-\-2ry-\-^y  is  also 
equal  to  2;ri-H- 2/j>  + fluxion  of  2xy.  Hence  fluxion 
of  2xy  =  2xy  +  2;ir;/. 

**  In  the  same  manner  as  the  quantities  ;r,  j/,  2,  are 
conceived  to  flow,  and  to  have  their  fluxions,  so 
may  the  quantities  i-,  j>,  i-,  be  supposed  to  be  variable, 
and  therefore  have  their  fluxions,  which  are  thus 
represented  x,y,  z,  and  are  called  the  second  fluxions 
of  x,y,  z"  (p.  II).  **The  fluent  of  any  quantity  as 
x"'x  is  represented  thus  \x'"x\.'^ 

William    West,   1762 

176.  William  West's  Mathematics'^  is  a  posthum- 
ous  work  ;  the  author  died  in  1760.  Fluxions  are 
treated  from  the  earlier  Newtonian  standpoint, 
infinitely  little  quantities  being  used.  Some  novelty 
is  claimed  for  this  text  in  the  treatment  of  maxima 
and  minima. 

Janics    Wilson^    1761 

177.  In  1761  Wilson  coUected  some  of  Benjamin 
Robins's  mathematical  tracts  in  a  two-volume  hook, 

1  Mathematics.  By  the  late  Rev.  Mr.  Wm.  West  of  Exeter.  Revised 
by  John  Rowe,  London,  1762.  There  appeared  a  second,  corrected, 
edition  in  1763. 


TEXT-BOOKS  OF  MIDDLROF  CRNTURY  201 

entitled  Matheniatical  Tracts  of  the  late  Benjamin 
Robins.  In  an  Appendix,  Wilson  inserts  some 
matters  of  historical  interest  regarding  certain 
manuscripts  of  Newton  ;  Wilson  also  defends  Robins 
against  criticisms  passed  by  a  French  writer,  and 
States  his  views  of  Maclaurin's  indebtedness  to 
Robins. 

Newton's  Method  of  Fluxions  (see  our  §  149)  was 
brought  out  in  Paris  in  1740  by  George  Louis  Le 
Clerk,  Comte  de  Buffon,  under  the  title,  La  méthode 
des  Fluxions^  et  des  suites  infiìiies.  Buffon  prepared 
a  historical  Preface,  in  which  he  criticised  severely 
Berkeley  and  Robins  for  presUming  to  take  excep- 
tion  to  anything  Newton  had  written  on  fluxions  or 
to  modify  Newton's  mode  of  exposition.  Buffon 
praises  Jurin,  and  then  speaks  of  Robins  thus 
(pp.    xxvii-xxix)  : 

''.  .  .  il  commence  par  le  censurer  &  par  dés- 
aprouver  sa  maniere  trop  brève  de  présenter  les 
choses  ;  ensuite  il  donne  des  explications  de  sa 
facon,  &  ne  craint  pas  de  substituer  ses  notions 
incomplettes  aux  Démonstrations  de  ce  grand 
homme.  Il  avoué  que  la  Geometrie  de  l'Infini 
est  une  science  certaine,  fondée  sur  des  principes 
d'une  vérité  sùre,  mais  enveloppée,  &  qui  selon  lui 
n'a  jamais  été  bien  connué  ;  Newton  n'a  pas  bien  lù 
les  Anciens  Géometres,  son  Lemme  de  la  Méthode 
des  Fluxions  est  obscur  &  mal  exprimé  ...  : 
malheursement  les  Mathematiciens  ont  été  plus 
incrédules  que  jamais,  il  n'y  a  pas  eu  moyen  de  leur 
faire  croire  un  seul  mot  de  tout  cela,  de  sorte  que 


204  LIMITS    AND   FLUXIONS 

Philalethes  comme  défenseur  de  la  vérité,  s'est 
chargé  de  lui  signifier  qu'on  n'en  croyoit  rien, 
qu'on  entendoit  fort  bien  Newton  sans  Robin,  que 
les  pensées  &  les  expressìons  de  ce  grand  Philo- 
sophes  sont  justes  &  très-claires  .  .  .,  ce  sont  des 
piéces  d'une  mauvaise  critique.    ..." 

Buffon  presents  no  argument  against  the  views 
expressed  by  Robins,  but  abuses  him  for  presum- 
ing  to  think  independently.  This  doting  attitude 
toward  Newton  is  justly  attacked  by  James  Wilson, 
in  bis  Appendix  to  the  Mathematica/  Tracts  of  the 
late  Benjamin  Robins^  voi.  ii,  London,  1761,  pp. 
325-327.  Wilson  rightly  says  that  if  it  was  a 
crime  for  Robins  to  make  mention  of  the  great 
brevity  with  which  Sir  Isaac  Newton  wrote,  Robins 
was  foUowed  in  it  by  Maclaurin  and  Saunderson. 
"The  truth  is,"  says  Wilson,  "Sir  Isaac  Newton 
at  first  made  the  same  use  of  indivisibles,  others 
had  done  :  in  his  Analysis  pei'  cequationes  numero 
terminorum  infinitas,  he  expressly  says,  *  Nec  vereor 
loqui  de  unitate  in  punctis,  sive  lineis  infinite 
parvis  ^  ;  '  and  in  his  Lectiones  Opticce  he  demon- 
strated  by  indivisibles."  Wilson  contends  further- 
more  that  Buffon  is  wrong  in  claiming  that  the 
mathematicians  paid  no  regard  to  what  Robins 
had  said,  that  in  fact  "the  best  writers  soon 
after  trod  in  Mr.  Robins's  steps."  In  fairness 
to  Buffon  it  should  be  said,  however,  that  he 
printed  his  Preface  in  1740,  and  that  Maclaurin, 
Saunderson,  de  Bougainville,  and  d'Alembert,  whom 
1  Cojnm.  Epist,  p.  85. 


TEXr-BOOKS  OF MIDDLEOF CENTUR  F205 

Wilson   mentions   as  foUowing    Robins,   wrote  at  a 
later  date. 

178.  James  Wilson  claims  ^  that  Maclaurin  in 
his  Fluxions  "  conformed  himself  entirely  to  Mr. 
Robins's  sentiments  in  regard  to  Sir  Isaac  Newton's 
doctrine,"  and  ''has  even  expressly  followed  his 
pian  in  treating  the  subject.  "  Jurin  had  contended 
(says  Wilson)  "that  Sir  Isaac  Newton's  method,  by 
proving  the  varying  quantities  carne  up  to  their 
limits,  was  more  perfect  than  that  of  the  ancients. 
Whereas  Sir  Isaac  Newton  never  claimed  such 
superiority  ;  .  .  .  The  coincidence  contended  for, 
and  thus  highly  praised  by  Philalethes,  is  the  very 
essence  of  indivisibles.  "  Wilson  rightly  insists  that 
Buffon 's  criticisms  of  Robins  are  unfair.  '*When 
he  talks  of  the  obscurity  of  Mr.  Robins's  ideas,  the 
insignificancy  of  his  phrases,  and  the  unintelligible- 
ness  of  his  style  ;  he  gives  the  most  certain  proof, 
that  he  had  never  carefully  read  his  writings,  .  .  . 
for  Mr.  Robins  is  much  admired  bere  for  the  con- 
trary  excellencies,  on  whatever  subjects  he  has 
employed  his  pen." 

179.  Wilson  represents  Philalethes  (Jurin)  as 
championing  the  use  of  the  infinitely  little  and  of 
indivisibles.  This  is  putting  the  case  too  strongly. 
In  his  papers  against  Berkeley,  Jurin  uses  quantities 
infinitely  little.  But  toward  the  end  of  his  debate 
with  Robins  he  begins  to  disavow  them.  Never 
did  Jurin  use  indivisibles.      Few  eighteenth-century 

*  Mathematical  Tracts  of  the  late  Benjamin  Robins,  voi.  ii,  London, 
17Ó1,  pp.  312,  315,  320. 


2o6  LIMITS    AND   FLUXIONS 

writers  have  brought  out  as  distinctly  and  clearly 
as  has  Jurin  the  difference  between  infinitesimals 
as  variables,  and  indivisibles  ;  Jurin  disavowed  ali 
quantity  "fixed,  determinate,  invariable,  indivisible, 
less  than  any  finite  quantity  whatsoever,"  but  he 
usually  did  admit  somewhat  hazily  a  quantity 
''variable,  divisible,  that,  by  a  Constant  diminu- 
tion,  is  conceived  to  become  less  than  any  finite 
quantity  whatever,  and  at  last  to  vanish  into 
nothing." 

Remarks 

i8o.  None  of  the  works  mentioned  in  this  chapter 
are  great  works.  Those  of  William  Emerson  and 
Thomas  Simpson  were  the  best  and  the  ones  most 
vvidely  used.  The  first  edition  of  Simpson  is  of 
earlier  date  (1737). 


CHAPTER  Vili 

ROBERT  HEATH  AND  FRIENDS  OF  EMERSON  IN 
CONTROVERSY  WITH  JOHN  TURNER  AND 
FRIENDS  OF  SIMPSON 

i8i.  The  principals,  Simpson  and  Emerson,  do 
not  themselves  appear  in  this  controversy.  During 
the  period  of  this  debate,  Robert  Heath  was  editor 
of  The  Ladies'  Diary,  which  appeared  once  every 
year  as  an  ahiianac.  We  begin  with  one  of  his 
articles. 

Robe7-t  Heath ^  1746 

182.    In   an   article,    Of  the  Idea,  and  Nature  of 
Fliixions,^  Heath  says  : 

'  *  The  Distinction  betwixt  the  Increments  and 
Fluxions  of  Magnitudes,  has  been  this  ;  that  the 
former  approach  in  Ratio  infinitely  near  the  latter, 
so  that  their  Difference  is  unassignable.  .  .  .  What 
we  cali  the  Fluxions,  or  Velocities  of  Magnitudes, 
are  only  the  Fluxions  in  Chief,  or  in  Part,  with 
which  they  are  born  ;  the  Part  neglected  in  the 
Ratio  exactly  corresponding  with  what  is  rejected 
in  the  finite  Ratio  of  the  infinitely  small  Increments, 
which   is   therefore   the   same   as   the   Ratio  of  our 

^   The  Ladies  Diary  :  or,  the  IVoman^s  Almanack.for  1746. 
207 


2o8  LIMITS    AND   FLUXIONS 

Fluxions.  And  hence,  whether  we  cali  those  finite 
Ratios,  Pluxions,  or  Increments,  their  Idea,  Nature, 
and  Originai  appear  to  be  the  very  same  thing. 
For  ali  Things  are  relative.  ..." 

He  argues  that  while  we  consider  a  line  or  piane, 
generating  an  area  or  solid,  as  of  no  thickness  in  the 
mind,  in  our  notation  we  represent  them  as  of  unit 
thickness,  **and  consequently  each  Line  or  Piane 
should  be  express'd  by  ^  x  L,  and  o  x  P,  to  denote 
them  as  they  are  in  the  Mind.  But  L  x  ^  to  o, 
and  P  X  <?  to  0^  are  in  the  same  Ratio  with  L  to  i, 
and  P  to  I ,  by  equal  Division  by  o  ;  and  those  again 
in  the  same  Ratio  with  Li-  to  i-,  and  Yx  to  i',  by 
equal  Multiplication  by  i-,  for  the  Ratio  of  Fluxions. 
But,  this  finite  Notation  of  Line  or  Plane^  which  we 
consider  of  no  Breadth,  or  Thickness,  and  yet  denote 
by  Unity,  each,  at  the  same  Time,  makes  the  Practice 
and  our  Comprehension  disagree.  ...  So  that  it  will 
be  an  Error  to  conclude  that  the  Ratio  of  the 
Fluxions  of  Quantities  generated  by  the  Motion  of 
Lines,  or  Planes,  is  arrived  at  this  Way,  without 
the  previous  Consideration  of  an  Increment  ;  for  the 
very  Lines  and  Planes  must  be  Increments,  or  Some- 
things  next  to  Notkings  themselves,  before  they 
were  what  we  finitely  express  them  by  Notation, 
or  Quantities  could  never  increase  or  be  generated 
thereby  :  For  to  carry  a  Line  or  Piane  of  no  Breadth 
or  Thickness  forward,  is  the  same  in  Terms  as  to 
carry  Nothing  forward.  And  therefore  the  Dis- 
tinction  between  the  Ratio  of  Increments,  and  that 
of   Fluxions,   is  only  what   the   Conception  of   the 


ROBERT  HEATH  v.  JOHN    TURNER     209 

Thing  differs  from  that  of  its  Notation  in 
Practice.  .  .  .  Those  who  desire  further  Satisfaction 
as  to  the  Nature  of  Fluxions,  of  their  noble  Use 
and  transcendant  Excellence,  may  consult  Mr. 
Emerson's  Doctrine  of  the  whole  Art^  which  is  .  .  . 
the  best  of  any.  .  .  .  Those  writers  will  find  them- 
selves  mistaken,  who  pretended  to  derive  the  finite 
Ratios  of  Motion,or  Fluxions  producing  Magnitudes, 
without  the  previous  Consideration  of  Incretnents, 
which  include  the  very  Notion  of  what  a  Fluxion  is. 
This  some  have  attempted  by  multiplying  Quanti- 
ties  into  their  Velocity,  and  some  by  other  Means, 
the  Result  of  which  originally  depends  on  incre- 
mentai Principles,  if  they  would  consider  the  Matter 
as  far  as  it  will  go.''  The  paper  is  brought  to  a 
finish  in  the  Ladies'  Diary  for  1747. 

Main  Articles  in  the  Controversy 

183.  Over  the  pseudonym  of  "  Cantabrigiensis  " 
there  appeared  in  1750  an  un  friendly  review  of 
Simpson's  Doctrine  and  Application  of  Fluxions.^ 
The  reviewer  contended  that  the  definition  of  a 
fluxion  as  the  ''magnitude  by  which  any  flowing 
quantity  would  be  uniformly  increased  "  (see  our 
§  172)  is  very  "  odd  "  ;  for,  *'in  quantities  uni- 
formly generated,  the  fluxion  must  be  the  fluent 
itself,  or  else  a  part  of  it."  Simpson's  endeavour 
to  exclude  *' velocity  "  "  cannot  be  made  intelligible 
without  introducing  velocity  into  it.  "      '' Again,  he 

^  Monthly  Review  ;    or,   New  Literary  Journal,   voi.   iv,    London, 
1750,  pp.  129-131. 

14 


2 IO  LIMITS    AND   FLUXIONS 

mistakes  the  efifect  for  the  cause  ;  for  the  thing 
generateci  must  owe  its  existence  to  something,  and 
this  can  only  be  the  velocity  of  its  motion  ;  but  it 
can  never  be  the  cause  of  itself,  as  his  definition 
would  erroneously  suggest.  "  Moreover,  it  is  strange 
that  Simpson  "  should  stili  stick  in  the  mud  and 
run  himself  into  the  old  exploded  method  used  by 
foreigners  ;  and  which  is  subject  to  ali  the  cavils 
that  bave  ever  been  raised  against  that  science." 

184.  This  criticism  originated  a  small  tempest. 
In  a  journal  called  Mathematical  Exercises,^  its 
editor,  John  Turner,  makes  certain  '*  Observations 
on  certain  invidious  Aspersions  on  Mr.  Simpson's 
Doctrine  and  Application  of  Fluxions,  published  in 
the  Monthly  Review  for  December  last,  by  Canta- 
brigiensis."  Mr.  Simpson  is  there  charged  as  having 
*'  mistaken  the  Effect  for  the  Cause  "  ;  Mr.  Simpson, 
says    Turner,    "  builds    upon    his    own    Definition; 

*  Mathematical  Exercises  No.  Ili  (i75i)>  P-  34-  Six  numbers  of  this 
journal  appeared  in  London  in  1750-1752.  No.  V  bears  the  date  1752  ; 
No.  VI  has  no  date.  Readers  are  invited  "  to  send  their  Performances 
(whether  new  Problems,  Paradoxes,  Solutions,  etc.)  Post  paid,  to 
be  left  with  Mr.  James  Morgan,  at  the  Three-Cranes,  in  Thames- 
street  .  .  ."  In  this  connection  a  statement  made  by  Charles  Hutton, 
in  his  Memoirs  of  the  Life  and  Writings  of  the  Author  [Thomas 
Simpson],  printed  in  Thomas  Simpson's  Select  Exercises  in  the 
MathematicSy  new  edition,  London,  1792,  p.  xviii,  is  of  interest  : 

"  It  has  also  been  commonly  supposed  that  he  [Thomas  Simpson] 
was  the  real  editor  of,  or  had  a  principal  share  in,  two  other  periodica! 
Works  of  a  miscellaneous  mathematical  nature  ;  viz.  the  Mathematician , 
and  Turner's  Mathematical  Exercises^  two  volumes,  in  8vo,  which 
carne  out  in  periodica!  numbers,  in  the  years  1750  and  1751,  etc.  The 
latter  of  these  seems  especially  to  bave  been  set  on  foot  to  afford  a 
proper  place  for  exposing  the  errors  and  absurdities  of  Mr.  Robert  Heath, 
the  then  conductor  of  the  Ladies  Diaryaxid  the  Palladium  ;  and  which 
controversy  between  them  endcd  in  the  disgrace  of  Mr.  Heath,  and 
expulsiou  trom  his  office  nf  editor  to  the  Ladies^  Diary,  and  the  substi- 
tution  of  Mr.  Simpson  in  his  stcad,  in  the  year  1753." 


ROBERT  HEATH  v.  JOHN    TURNER    211 

which,  he  tells  us,  himself,  is  not  exactly  the  same 
as  that  of  Sir  Isaac  Newton."  Mr.  Simpson  is  also 
charged  with  plagiarism  from  Cotes's  Estimatio 
Errorum.     John  Turner  says  : 

"  Here  his  Remarks  on  the  Author's  Definition 
of  a  Fluxion  first  demand  our  Consideration  :  Mr. 
Simpson  makes  it  to  be,  *  the  Magnitude  by  which 
a  flowing  Quantity  would  be  uniformly  increased  in  a 
given  Time.'  This  Definition  the  Critic  represents 
as  a  very  old  one  ;  and  with  regard  thereto  advances 
the  two  following,  extraordinary,  Positions  : 

'*  I.  That,  in  Ouantities  uniformly  generated,  the 
Fluxion  must  (according  to  the  said  Definition)  be 
the  Fluent  itself,  or  else  a  Part  of  it. 

**2.  And  that,  in  other  Quantities  generated  by 
a  variable  Law,  the  Fluxion  will  not  be  a  real,  but 
an  imaginary  Thing. 

'*To  the  first  of  these  Objections  I  answer,  that 
the  Fluxion  is  neither  the  Fluent  itself  nor  a  Part 
of  it  :  it  is  a  Quantity  of  the  same  Kind  with  the 
Fluent  ;  but  the  Fluent  being  the  Quantity  already 
produced  by  the  generating  Point,  Line  or  Surface, 
supposed  stili  in  Motion,  and  the  Fluxion  what  will 
arise,  hereafter,  from  the  Continuation  of  that 
Motion  ;  the  latter  can  no  more  be  denominated 
a  Part  of  the  former  than  the  ensuing  Hour  a  Part 
of  the  Time  past. 

"But  his  second  Observation  is  a  stili  more 
glaring  Instance  of  his  Disingenuity,  and  Want  of 
Judgment.  Does  it  follow,  because  a  Body,  really, 
moves  over  a  certain   Distance,    in   a  given  Time, 


212  LIMITS    AND   FLUXIONS 

with  an  accelerateci,  or  a  retarded  Velocity,  that 
there  is  no  Distance  over  which  it  might  pass  in 
the  same  Time,  with  its  first  Velocity  uniformly 
continued.  The  Space  over  which  a  Body  would 
uniformly  move  with  such,  or  such,  a  proposed 
Velocity,  is  no  less  real  because  no  Part  of  it  is 
actually  described  with  that  Velocity  "  (pp.  36,  37). 

185.  Then  foUows  an  article  reprinted  from  the 
Daily  Gazetteeriox  December  4,  last  [1750],  in  which 
one  who  signs  himself  ''Honestus"  (said  to  have 
been  John  Turner  himself)  charges  that  the  compiler 
of  the  Ladies'  Diary  (Robert  Heath)  is  also  the 
compiler  of  the  Palladium,  and  the  best  material 
designed  by  contributors  for  the  Diary  are  reserved 
by  him  for  the  Palladium  ;  that  the  latter  publica- 
tion  is  owned  by  the  compiler,  while  the  former  is 
not.  Robert  Heath  wrote  a  reply  in  the  Daily 
Gazetteer  of  December  6  ;  four  letters  follow  on 
this  subject. 

186.  John  Turner's  defence  of  Simpson  led  to  the 
publication  of  what  Turner  called  a  ''scurrilous 
Pamphlet."  This  pamphlet  is  without  doubt  the 
TrutJi  Triuiìiphant :  or  Fluxions  for  the  Ladies,^ 
London,  1752,  or  else  those  parts  in  that  pamphlet 
which  appear  over    the   pseudonyms   *' X  Primus  " 


^  The  fuller  title  of  the  pamphlet  is  thus  :  Trttih  Triumphant  :  or, 
Fluxions  for  the  Ladies.  Shewing  the  Cause  to  be  before  the  Effect^  and 
different  from  it  ;  That  Space  is  not  Speed^  nor  Magnitude  Motion. 
With  a  Philosophic  Vision,  Most  humbly  dedicated  to  his  Illustrious 
High  and  Serene  Excellence,  the  Sutt.  For  the  Information  of  the 
Public^  by  X,  F,  and  ^,  who  are  not  of  the  Family  of  jt,  y,  z,  biit 
near  Relations  of  x\  y\  and  z  .  .  .  .  London.  Printed  for  W.  Owen, 
M.DCC.LII. 


ROBERT  HEATH  v.  JOHN    TURNER     213 

and  **Y  Secundus. "  These  documents  evidently 
emanated  from  the  pen  of  Robert  Heath,  assìsted 
possibly  by  some  other  adherents  of  William 
Emerson.  At  the  risk,  perhaps,  of  not  observing 
strict  chronological  sequence,  we  proceed  to  the 
consideration  of  ali  parts  of  TrutJi  TriinupJiant. 
In  the  dedication  ''to  the  Sun,"  it  is  stated  that 
"the  Family  of  the  Wou'd-be's  in  this  Island  is 
become  very  numerous,  by  uniforvily  continuing  in 
their  Errors."  Thus,  both  the  title-page  and  the 
dedication  play  on  Simpson's  definition  of  fluxions 
and  its  alleged  defects.  In  the  Preface  one  reads  : 
"Fluxions,  then,  Ladies,  that  have  so  puzzled  our 
wise  Mathematicians  to  define,  are  the  respective 
Degrees  of  Motion,  at  any  Instant  of  Time,  of  any 
two  things  or  Bodies  that  continually  flow,  or  move 
on,  over  Space."  Four  pages  are  devoted  to  the 
explanation  of  fluxions. 

187.  Then  follow  the  two  criticisms  of  John 
Turner's  defence  of  Simpson,  signed  "X  Primus  " 
and  "Y  Secundus,"  to  which  we  have  alluded  above. 
In  the  former  of  these  articles  John  Turner  is 
treated  with  contempt.  "  Who  this  John  Turner 
is,  whether  he  is  Mr.  Simpson's  Clerk,  or  his  Pupil, 
or  some  Dependant  on  him  ;  or  whether  he  be 
Mr.  Simpson  himself,  is  not  very  material  to  the 
Reader  ..."  Turner  is  continually  referred  to 
as  "John."  To  Turner's  reply  to  the  first  criticism 
on  Simpson's  text,  "X  Primus"  makes  rejoinder  : 
''''John  says,  the  Pluent  being  the  Ouantity  already 
produced — Pray  how  was  this   Ouantity  produced. 


214  LIMITS    AND   FLUXIONS 

by  some  magic  Art,  without  any  Fluxion  ?  I  believe 
not.  .  .  .  For  my  Part,  l  know  of  no  Body  that 
ever  saìd,  that  the  Parts  of  the  Fluent  that  went 
before  were  generated  by  the  Fluxion  that  is  to  come 
after,  but  every  Part  by  its  proper  Fluxion.  ..." 
To  Turner's  reply  to  the  second  criticism,  ''X 
Primus  "  makes  rejoinder  : 

'*  If  there  be  no  Magnitude  by  which  the  flowing 
Ouantity  is  really  increased,  such  a  Magnitude  is  not 
real,  but  an  imaginary  Thing  only  .  .  .  But  John 
thinks,  that  every  Thing  that  exists  in  his  Imagina- 
tion,  really  exists  in  Nature  .  .  .  Sir  Isaac  Newton 
defines  PTuxions  by  the  Velocities  of  the  Motions. 
But  Mr.  Simpson  declares  against  this,  and  likewise 
tells  US,  that  by  taking  Fluxions  for  mere  Velocities, 
the  Imagination  is  confin'd,  as  it  were,  to  a  Point. 
How  /lis  Imagination  is  confin'd  I  don't  know  ;  but 
Sir  Isaac  Newton  chused  to  define  it  thus,  as  very 
well  knowing,  that  this  is  the  only  soà'd  Foundation 
upon  which  it  could  be  defended  against  ali  the 
impertinent  Cavils  of  ignorant  or  weak  Pretenders. " 
The  parting  shot  by  "  X  Primus"  is — your  Great 
Master  will  not  ^'think  you  a  fit  Champion  to 
engagé  in  his  Cause  for  the  future  ;  so,  good  Night, 
John." 

i88.  The  reply  made  by  '*  Y  Secundus  "  is  to  the 
effect  that  the  defender  of  Simpson  is  ''equally  in 
the  Dark"  with  Simpson  himself,  ''otherwise  he 
would  not  have  gone  about  to  defend  so  defenceless 
a  Cause,  as  to  vindicate  an  Absurdity,  by  repre- 
senting  a  Pluxion  to  be  of  the  same  Kind  with  the 


ROBERT  HEATH  v.  JOHN    TURNER     215 

Fluente  uniformly  generateci  ;  when  the  one  is  a 
Ouantity  of  actual  Velocity,  and  the  other  a  Ouantity 
of  Space,  described  by  that  Velocity,  which  can  be 
only  proportional  to  it." 

189.  After  some  poetry  **To  the  Family  of  the 
the  Wou'd-be's,"  foUovv  "  Animadversions  on  Mr. 
Simpson's  Fluxions,"  "  By  2  Tertius,"  vvho  quotes 
a  criticism  of  Simpson  from  the  pen  of  J.  Landen. 
Where  Landen's  review  first  appeared  we  do  not 
know.  As  quoted  here,  Landen  objects  to  the 
definition  of  fluxions  ^  '  diS  faulty,  by  the  Author's 
difìferent  Idea  given  of  them  to  that  by  the 
Inventor";  Landen  disapproves  of  *'denoting  ali 
Ouantities  whatsoever  by  Lines,  to  bring  them  to 
one  Denomination,  and  those  Lines,  to  be  described 
by  Bodies  in  Motion."  In  criticism  of  fluxions  in 
general,  Landen  says  that  the  finding,  from  the 
velocities,  the  spaces  passed  over,  and  vice  versa, 
"  may  be'  managed  by  common  Algebra,  without 
the  least  Obscurity.  The  Business  had  always 
been  better  considered  in  that  Light,  without  ever 
making  Use  of  the  Term  Fluxions,  as  if  a  new  Kind 
of  Analysis,  tho',  in  Faci,  only  the  Do  e  trine  of 
Motion  improved,  and  applied  to  Purposes  before 
unthought  of.  " 

190.  The  next  article  in  Truth  Triumphant  is  a 
reprint  of  the  first  criticism  of  Simpson,  contributed 
in  1750  by  Cantabrigiensis  to  the  MontJily  Review. 
Eight  more  articles  concerning  motion,  fluxions,  and 
mechanics  bring  the  pamphlet  to  a  dose  ;  they 
make  no  reference  to  Simpson,      '' Heliocentricus  " 


2i6  LIMITS    AND   FLUXIONS 

explains  higher  fluxions  in  a  way  that  cannot  be 
called  illuminating. 

191.  Then  "  Amicus  "  speaks  '*  of  the  Use  of  the 
Algebraic  Cypher,  in  finding  the  Fluxions  of  Alge- 
braic  Ouantities,"  letting  x  increase  or  decrease,  and 
become  ;fzb^,  where  the  *' Increments  or  Decre- 
ments  are  seen  to  be  ±<^,"  and  **dividing  by 
algebraic  0"  thereupon  "  algebraically  considering 
0  of  insensible  Value,  as  before  it  was  consider'd  of 
real  sensible  Value."  Taking  a  reminiscent  mental 
attitude,   *'  Amicus  "  says  : 

'  '  This  algebraic  Ratio  of  the  Fluxions  of  Quantities, 
to  which  the  diminishing  Value  of  the  algebraic 
Increments  or  Decrements,  from  their  limited  State 
or  Value,  tend  together,  to  their  geometrical  vanish- 
ing  (by  supposing  the  variable  Value  less  and  less) 
has  been  misconceived,  as  vanishing  together  with 
the  ;t^/ geometrical  Increments  or  Decrements  they 
are  the  Value  of\  whence  o  has  been  denominated 
a  departed^  instead  of  an  algebraic  Quantity,  by  a 
famous  B— /,  tho'  it's  Reality  and  Presence  stili 
existed  before  his  Eyes  ;  but  if  0^  the  Cypher- Value, 
or  algebraic  Quantity,  call'd  also  Nothing^  be  made 
to  signify  Nothing,  because  it  is  so  call'd,  the  Word 
Nothing  with  as  much  Propriety,  may  be  called  no 
Word,  be  allowed  to  bave  no  Signification  among 
other  Words,  and  be  deem'd  a  mere  Blank,  as  no 
Subject  capable  of  Consideration."  Further  on  in 
the  pamphlet,  the  query  whether  there  can  be  *'  real 
Motion  in  no  Time,"  for  ''any  one  Point  of  inter- 
mediate Space  gone  over;  especially  since  an  infinite 


ROBERT  HEATH  v.  JOHN    TURNER     217 

Number  of  Points  can  never  actually  constitute  real 
Magnitude,"  and  whether  '' Motion,  or  Fluxion, 
can  aciually  exist,  and  be  known,  but  by  the  next 
Increment  of  Space  gone  over,  in  some  real  and 
next  Monnent  of  Time"?  These  are  fundamental 
problems  indeed.  Zeno  is  not  mentioned  in  the 
pamphlet,  but  the  query  involves  Zeno's  subtle 
paradox  of  the  '  '  arrow.  "  Nor  is  the  answer  given 
devoid  of  interest.  '*  But  Time,  and  Motion,  flow- 
ing  over  Space,  .  .  .  {since  no  Quantity  can  be 
assign'd,  or  imagin'd  so  smalla  but  there  will  stili  be 
smaller)  the  respective  Degree  of  Velocity  of  Motion, 
or  Fluxion  (i.e.  instantaneous  velocities)  of  that 
Flowing  at  any  Instant  of  Time,  and  Point  of  Space 
gone  over,  will  be  everywhere  assignable  by  the 
immediate  Increments,  as  Effects  of  those  preceding 
Velocities,  as  has  been  shewn.  Whence  it  will 
follow,  that  certain  Degrees  of  Motion,  Fluxion,  or 
Velocity,  exist  at  every  instant  of  Time  taken,  and 
Point  of  Space  respectively  described  ;  contrary  to 
the  dijfferential  Notion  that  P^oreigners  have  of  this 
Matter.  "  The  weak  spot  here  resides  in  the  words 
''immediate  Increments";  do  immediate  increments 
exist  in  view  of  the  statement  in  the  above  paren- 
thesis  ?  The  lack  of  a  satisfactory  arithmetical  con- 
tinuum Comes  to  view  more  fuUy  in  the  antagonism 
between  geometrie  increases  and  algebraic  increases 
exhibited  in  the  following  passage  taken  from  the 
pamphlet  : 

"AH    the    Values  of  the   geometrical    Increases 
flow'd  over,  in  finite  Time,  can  never  be  algebraically 


2i8  LIMITS    AND   FLUXIONS 

express'd  in  infinite  Time  ;  in  which  Sense  the 
algebraic  Increases  being  again  diminish'd,  are  said 
never  to  converge  to  the  Limits  of  their  geometrical 
Magnitudes  in  Motion,  but  will  stili  have  sensible 
Value  ;  yet  supposing  the  geometrie  Increases,  and 
their  algebraic  Values  to  flow  and  decrease  alike, 
to  something  determinate,  then  o^  and  <?,  and  it's 
Powers  into  a  Variable  Quantity,  and  it's  Powers, 
will  accurately  express  the  Limits  of  variable  Quan- 
tities,  or  Beginnings  of  their  Increases  ;  which 
Limits^  or  Beginnings  of  Increases  of  Ouantities, 
are  accurately,  as  the  Fluxions  of  those  Ouantities 
in  general." 

"  Visionarius  "  closes  with  a  philosophic  vision  in 
which  four  candidates  for  honors  appear  before  the 
Goddess  of  Science.  Rejected  are  the  first  three, 
viz.  the  author  of  M atheinatical  Exercises  (John 
Turner),  the  one  holding  to  the  motto  **  A  cypher 
is  no  Algebraic  Quantity  "  (Bishop  Berkeley),  and 
a  Grand  Magnifier  of  Fluxions  (Thomas  Simpson)  ; 
crowned  is  the  author  of  the  incomparable  treatises 
of  Fluxions  and  Trigonometry  (William  Emerson), 
in  whose  interests  Truth  Triuniphant  appears  to  be 
mainly  written. 

192.  The  probability  is  that  the  **scurrilous 
pamphlet  "  to  which  John  Turner  made  reply  in 
an  issue  of  his  Mathematical  Exercises  was  only  a 
part  of  what  is  given  in  Truth  Triuniphant.  The 
latter  is  probably  a  later  and  enlarged  publication. 
In  that  reply  Turner  argues  that  **  it  must  appear  to 
everyone  that,  what  Mr.  Simpson  defines  as  Fluxions, 


ROBERT  HEATH  v.  JOHN    TURNER     219 

are  exactly  such  finite  Ouantities,  proportional  to 
the  Velocities  as  Sir  Isaac  Newton  here  ^  speaks  of  ; 
since  it  is  well  known  that  the  Quantities  produced, 
or  the  Distances  described,  in  any  given  Time,  by 
Motions  uniformly  continued,  are,  accurately,  as 
the  Velocities  of  the  said   Motions." 

193.  In  No.  V  (1752)  of  the  Mathematical 
Exercises  appears  another  article  in  the  contro- 
versy,  written  by  John  Turner.  There  is  little  in 
it  requiring  our  attention.  It  is  a  reply  to  two 
pamphlets,  the  Lady's  Philosopher  and  a  new 
Pal/admm,  both  publications  from  ,  the  pen  of 
Robert   Heath. 

Ladies'  Diary^   175 1,   1752 

194.  The  Ladies'  Diary  iox  175  i  has  an  article  on 
The  Nature  and  Use  of  the  Algebraic  Cypher^  or 
Quantity  o,  *'  by  Fluxioniensis  "  ;  o  /  o  is  proved  to 
signify  ''any  Value  at  Pleasure  by  considering 
{cT  —  X'') ^ {a  —  x)  {orn=  I,  2,  3,  4,  etc. ,  when  x  =  a.'' 
This  "confutes  the  Notion  of  some  Mathematicians" 
that  o  /  o  expresses  "  a  Ratio  of  Equality."  Next 
it  is  argued  that  o°=i.  "  Hence,"  says  a  second 
anonymous  critic,  "ali  Cypher-Paradoxes,  and 
Mysteries  of  Ultimate  Ratios,  or  Ratios  of  Least 
Increments  or  Decrements  of  Quantities,  vanish  and 
Day  appears.   ..." 

"  Waltoniensis,  making  a  Distinction  between 
0  signifying  some  Quantity,  and  o  signifying  no 
Quantity,   or  absolute  Nothing,  says   that  when  x 

*  Principia^  Bk.  II,  Lemma  2.     See  our  §§  16-19. 


220  LIMITS   AND   FLUXIONS 

converges  to  o,  in  the  ultimate  State  before  it 
vanishes,  x^'^i  \  but  says,  when,r  entirely  vanishes, 
or  becomes  absolutely  of  no  Value,  that  then 
x^  —  o^  =  o\  But  0  being  supposed  no  Ouantity  is 
contradicted  by  Algebraic  Computation,  which  is 
general  and  retains  o,  in  a  Mathematica!  Sense,  for 
a  Ouantity  in  the  Scale,  as  much  as  any  other 
Figure  or  Literal  by  which  Ouantity  is  denoted  and 
compared.  .  .  .  Waltoniensis  farther  observes  that 
Fluxions  are  the  Limits  to  which  the  Ratios  of  the 
Increments  or  Decrements  of  Quantities  converge, 
and  are  assignable  from  the  Principles  of  Motion 
only  {uniform^  accelerateci^  and  retarded)  ;  and  thinks 
the  Doctrine  has  nothing  to  do  with  infinitely  small 
Quantities,  First  and  Last  Ratios  ;  and  that  only 
finite  Quantities  need  be  introduced — '  to  avoid 
Disputes,  and  the  dark  Mists  spread  over  the  Pro- 
cess,  different  to  the  demonstrative  Lights  of  the 
Antients.'  But  Motion  refers  to  the  Spaces  passed 
over,  by  which  it  is  comprehended,  measured,  and 
compared  :  And  tho'  Mr.  Simpson  has  pretended  to 
deduce  the  Ratios  of  Fluxions  of  Quantities  without 
the  use  of  indefinitely  small  Quantities  (see  his  New 
Doctrine  and  Application  of  Fluxions)  yet  the  Motion 
of  his  Points  along  the  Lines  answers  to  them  by 
the  indefinitely  small  Spaces  described  together, 
and  are  to  the  same  Effect  as  Quantities  taken  in- 
definitely small  ;  which  Sir  Isaac  Newton  himself 
introduced  to  illustrate  the  Quantity  of  relative 
Motion  by.  Fluxions^  as  instantaneous  Velocities, 
are  also  as   the   Increases  or   Quantities  of  Space 


ROBERT  HEATH  v.  JOHN    TURNER     221 

passed  over  together  by  those  instantaneous  Veloci- 
ties^  uniformly  continued  ;  but  are  not  the  Spaces  or 
Ouantities  themselves  that  would  be  described  by 
them  according  to  Mr.  SimpsorCs  new  Theory,  and 
Application.  See  Emerson' s  Doctrine  and  Applica- 
tion of  Fluxions.      Price  6s.  only." 

195.  In  the  Ladies'  Diary  for  1752  the  reader 
is  amused  by  satirical  remarks  on  mathematicians. 
There  is  also  a  continuation  of  the  discussion  **  Of 
the  Cypher-Value  and  Office  of  the  Algebraic 
Quantity  ^."  *' Nihil  Maximus  says  that  -9999, 
etc.  ad  inf.  will  never  converge  to  i,  nor  yet 
I  /  (10,000,  etc.  ad  inf.)  to  o\  because  any  Quantity 
infinitely  increased  or  diminished  will  be  stili  greater 
or  less,  and  never  numerically  arrive  at  Infinity,  and 
0  Value.  .  .  .  That  a  Distinction  should  be  care- 
fully  made  between  what  are  called  infinite  and 
indefinite  great  and  small  Quantities  {the  former  of 
which  being  impossible)  ;  for  what  is  of  indefinite 
Value  has  Equality,  tho'  it  may  be  sometimes  un- 
assignable  ;  while  what  is  infinite  is  never  determin- 
able,  and  has  never  Equality.  Hence  the  numerical 
Value  of  0  and  i  /  (1000,  etc),  will  be  for  ever 
different  ;  one  being  a  Quantity  of  no  sensible 
Value,  but  yet  significant,  and  the  other  of  indefi- 
nite small  Value.  .  .  .  Infinite  Quantity,  or  in- 
finite numerical  Value,  expressed  by  Authors,  is 
neither  practicable  nor  comprehensible.  .  .  .  An 
infinite  Series  can  never  precisely  converge  to  a 
finite  or  determinate  Value  ;  because  it  for  ever  ì-uns 
on.     The  finite  Value,  taken  for  that  of  an  infinite 


222  LIMITS    AND   FLUXIONS 

Series,  is  only  the  Value  from  whence  that  infinite 
Series  is  or  may  be  derived.  Mr.  Landen  thinks 
that  the  Value  o  is  no  algebraic  Ouantity  ;  but  calls 
it  a  mere  Blank^  or  absolute  Nothing  ...  ;  he 
says,  that  its  peculiar  Office  is  only  in  arithmetic 
Notation  ;  while  we  see  it  applied  to  other  Use  and 
Office  in  Treatises  of  Algebra  and  Fluxions,  as 
also  by  himself,  for  an  algebraic  Character  or 
Quantity  o^   its  own  Value." 

This  discussion  of  o  is  continued  at  great  length. 
Confusion  arises  from  the  doublé  use  of  the  symbol 
and  from  the  difficulties  surrounding  o  as  the  limit 
of  variables  or  sequences.  Reference  is  made  twice 
in  this  Ladies'  Diary  (1752)  to  the  pamphlet  Trutk 
Triuinphant,  or  Fluxions  for  the  Ladies,  where  the 
nature  and  office  of  0  are  discussed,  and  exception 
is  taken  to  Landen's  views  on  ^.  *'  Fluxioniensis  " 
says:  '*  And  therefore  I  should  not  stick  to  rank 
this  excellent  Reasoner  with  the  great  Master  of 
Reason  he  mentions,  the  B — p  of  Cloyne,  as  he 
clearly  appears  to  be  of  the  same  class.  " 

Poptilar  Intprcssion  of  the  Nature  of  Fluxions 

196.  A  reviewer  of  Richard  Jack's  Euclid's  Data 
Restored^  quotes  from  Jack's  preface  what  appears 
to  be  the  opinion  of  a  non-specialist  : 

"  Others,  who  claim  the  honour  of  extending 
their  principles,  treat  of  what  they  cali  Fluxions, 
calculus  differentialis,  infiniment  petifs^  extreme  and 

^  Monthly  Review^  voi.  xvi,  London,  1757. 


ROBERT  HEATH  v.  JOHN    TURNER     223 

ultimate  ratios,  etc,  and  with  so  much  obscurity, 
that  no  distinct  idea  of  the  thing  treated,  is  com- 
municated  to  the  mind  of  the  Reader.  From  their 
want  of  that  precision  and  perspecuity  which  the 
Ancients  carefully  observed  in  ali  their  writings, 
the  mind  becomes  clouded  with  confusion,  begins 
to  doubt,  which  terminates  in  a  disbelief  of  their 
principles  ;  for  which  reason  they  have  been  often 
called  upon  to  demonstrate  them  :  but  no  demon- 
stration  has  appeared." 

To  this  the  anonymous  reviewer  of  Jack's  hook 
replies  : 

"That  the  principles  of  Fluxions  stand  in  need 
of  demonstration,  especially  since  the  publication  of 
Maclaurin's  works,  is  certainly  a  mere  pretence,  made 
only  to  cover  the  ignorance  of  the  objector  ..." 

Remarks 

197.  In  this  chapter  we  have  given  views  held  by 
writers  representing  the  rank  and  file  of  mathe- 
matica! workers.  In  several  passages  the  need  of 
an  adequate  theory  of  a  linear  continuum  makes 
itself  strongly  felt. 

Some  curiosity  attaches  to  the  following  contem- 
poraneous  opinion  of  Tvuth  Triuiìipìiant  :  ^ 

*'This  is  an  odd  assemblage  of  controversial 
scraps,  chiefly  relating  to  some  disputes  concerning 
Mr  Emerson's  treatise  on  fluctions,  and  Mr.  Simp- 
son's  on  the  same  subject.      This  most  unimportant 

^  Montili)'  RevieWy  voi.  v,  London,  1751,  p.  462. 


224  LIMITS    AND   FLUXIONS 

controversy  was  first  occasioned  by  the  few  obser- 
vations  on  Mr  Simpson's  book  published  in  the 
Monthly  Review.  .  .  .  The  author  writes  in  a 
manner  that  can  do  little  honour  to  any  party  or 
opinion.  And  why  he  chose  to  give  this  strange 
and  insignificant  production  so  odd  a  title,  is  a 
mystery  that  none  but  himself  can  clear  up.  " 


CHAPTER  IX 
ABORTIVE  ATTEMPTS  AT  ARITHMETISATION 

John  Kirkby,    1748 

198.  In  the  preface  to  his  Doctrine  of  Ultirnators  ^ 
the  author  states  that  his  doctrine  **depends  upon 
scarce  any  Thing  else  but  a  due  Application  of  the 
Cypher  0  to  the  analogous  Office  in  Universal 
Arithmetic,  which  ìt  is  always  known  to  occupy  in 
Common  Arithmetic."  He  argues  that  -'the  super- 
lative impropriety  of  the  Word  Fluxion,  when 
applied  to  this  Purpose,  will  fully  appear  ;  when  we 
come  to  consider,  that  it  is  put  to  express  an  Idea, 
which  arises  from  the  Contemplation  of  Ouantities 
purely  as  Quantities  :  that  is,  in  the  same  abstract 
Manner,  as  they  are  the  proper  Subject  of  Algebra, 
exclusive  of  every  other  Considerati on  ;    and  con- 

^  The  Doctrine  of  Ultirnators.  Containing  a  new  Acquisition  to 
Mathematical  Literature,  naturally  resulting  from  the  Consideration  of 
an  Equation,  as  reducible  from  its  variable  to  its  ultimate  State  :  Or,  a 
Discovery  of  the  true  and  genuine  Foundation  of  what  has  hitherto  mis- 
takenly  prevailed  under  the  improper  Names  of  Fluxions  and  the  Differ- 
ential  Calculus.  By  means  of  which  we  now  have  that  Apex  of  ali 
Mathematical  Science  entirely  rescued  from  the  blind  and  ungeometrical 
Method  of  Deduction  which  it  has  hitherto  laboured  under  ;  andmade  to 
depend  upon  Principles  as  strici ly  demonstrable,  as  the  most  selfevident 
Proposition  in  the  first  Elements  of  Geometry.  By  the  Reverend  Mr. 
John  Kirkby,  Vicar  of  Waldershare  in  Reni.  London,  MDCCXLVIIl. 
Pp.  144. 

225  15 


226  LIMITS    AND   FLUXIONS 

sequently  have  not  the  least  Regard  to  Time  or 
Motion,  which  are  necessarily  implied  in  a  Fluxion. 
And  the  essential  Property  of  a  Fluxion  is  certainly 
excluded,  after  the  most  singular  Manner,  in  the 
Idea  of  Quantity  considered  at  its  Ne  plus  ultra  : 
that  is,  in  other  Terms,  when  it  is  in  a  State, 
where  ali  Possibility  of  such  imaginary  Flux  is 
taken  from  it.  So  that  the  Term  Fluxion,  when 
used  to  this  Purpose,  if  it  have  any  Meaning  at  ali, 
is  as  contrary  to  the  true,  as  Darkness  is  to  Light." 
He  takes  an  algebraic  equation  A^o±B^'it:C<3:2jt 
D<23±  .  .  .  ±Z<3:"  =  o,  assumes  the  coefficients  Y 
and  Z  of  the  two  highest  terms  as  fixed,  and 
declares  (without  proof)  that  the  absolute  term  A 
is  a  maximum  when  the  n  roots  of  the  equation  are 
equal.  When  such  an  equality  exists,  the  equation 
is  reduced  '*to  its  ultimate."  When  the  roots  are 
equal  he  represents  them  by  +<:  or  —e.  To  reduce 
the  trinomial  A±Brt±Z^  =0  to  its  ultimate,  "  we 
must  make  B  /  Z  =  -:^nc''~^  in  the  n  Power  of  ^zh^ 
=  o.  That  is  (because  ^=^)  B  /  Z.  =  na**~'^.  There- 
fore  the  Ultimate  required  is  ^■:^nZa''-'^  =  o,  or 
B^"zb^Z<2'*"  1^:0  =  0."  To  be  observed  here  is  that 
Kirkby  connects,  though  only  in  an  obscure  way, 
his  ultimate  with  the  coefficient  of  a  in  the  second 
term  of  the  binomial  expansion  of  {c-±:,ay\  He  then 
pretends  to  prove  ''that  the  Ultimate  of  the  Sum 
of  never  so  many  Equations  is  the  same  with  the 
Sum  of  their  respective  Ultimates  "  ;  hence,  the 
Ultimate  of  the  above  general  equation  is  f?±B^®it 
2Caa^-^lDa^a^-±z  •    •    •    ^nZa"~^a^  =  o.       He    gives 


ATTEMPTS   AT  ARITHMETISATION   227 

the  rule  for  finding  the  "ultimate"  or  "  ultimator  " 
of  a"  \  this  ultimator  is  ua'^'^a^;  he  also  gives  the 
rule  for  writing  down  '  '  the  subject  of  every  Ulti- 
mator"; the  subject  of  the  ultimator  na'^'^a^  h^ing 
a"  +  c.  He  applies  these  rules  when  the  exponents 
are  fractional. 

The  **  ultimator  "  of  the  product  of  two  variables, 
ae,  is  found  thus.  **Put  ae  —  bee^  and  ae  =  caa. 
Whence  a  —  be,  and  e  =  ca^  and  ae  =  \  bee  +  \  caa. 
The  Ultimator  of  which  last  is  .  .  .  bee^-\-caa^,  and 
consequently  is  equal  to  the  Ultimator  of  ne.  But 
be  =  (i^  and  ca  =  e.  Therefore  these  substituted  for 
their  Equals  in  that  Ultimator  give  ae^-^ea^  for  the 
Ultimator  of  ae''  (p.  43).  It  will,  of  course,  be 
noticed  that  special  limitations  are  placed  upon  the 
variables  a  and  e,  when  the  coefficients  b  and  e  are 
tacitly  assumed  to  be  constants.  Kirkby  proceeds 
to  the  derivation  of  the  ultimators  of  fractions  and 
logarithms.  He  explains  the  necessity  of  retaining 
in  the  Ultimator  each  variant  (variable)  under  its 
0  Power.  **  Without  this  we  cou'd  bave  no  Means 
from  the  Nature  of  the  thing  itself,  whereby  to 
distinguish  an  Ultimator  from  a  Subject."  The 
functions  of  a^,  e^  are  more  than  simply  to  represent 
unity  ;  just  what  they  are  is  not  very  clear,  although 
to  the  author  "  it  is  evident  then,  as  often  as  any 
Subject  consists  of  different  Variants  Ex  gr.  ,i',  j,  ^, 
that  the  Expressions  x^^  j®,  s^,-  in  the  Ultimator 
have  the  same  Difference  in  Power  with  the 
same  Variants  under  any  other  common  Exponent 
x'\  y\  ::*"....    Therefore  the  Expression  x^,  f^,  5*^, 


228 


LIMITS    AND   FLUXIONS 


I  conce i ve  may  be  each  fìtly  called  the  Peculiar  Unii 
of  its  respective  Scale  of  Powers.  Hence  every 
Ultimator  may  be  defined  to  be,  The  proper  Reference 
of  each  Subject  Ì7i  a  given  Equation  to  the  Peculiar 
Units  of  the  Powers  of  ali  its  VariantSy  in  Order  to 
discover  the  Ratios  of  those  Variants  to  one  another  in 
their  Ultimate  State.  Which  1  take  to  be  the  true 
Defìnition  of  what  has  been  hìtherto  most  impro- 
perly  and  unintelligibly  called  a 
Fluxion  by  some,  and  a  Differ- 
ential  by  others  "  (page  49). 

199.  Kirkby's  doctrine  may 
perhaps  become  plainer  by  the 
study  of  one  of  his  applications. 
In  any  curve  with  the  concave 
side  to  TO,  the  greater  abscissa 
VP  (or  v)  has  always  the  greater 
"semi-ordinate"  PM  (or  s), 
"and  each  are  the  greatest 
that  they  possibly  can  be  to 
the  same  Arch  VM,  or  to  the 
same  intercepted  axis  VR.  Therefore  the  Sub- 
normal  PR  (or  r  —  v),  and  consequently  the 
Normal  MR  (  =  c)  are  each  the  least  that  they 
possibly  can  be  to  the  same  Arch  VM,  or  the 
same  intercepted  Axis  VR  (or  r).  Therefore,  if 
in  the  last  Equation  [r^  —  c'^=2rv  —  v^—s^\  e  and 
r  be  invariable,  v^e  have  r^  —  c^  an  Ultimum,  Con- 
sequently, the  Ultimate  of  that  Equation  .  .  .  is 
2;'z;"  —  2vv°  —  2j^°  =  o, or  (dividing  by  2)  j^°  =  r  —  z^  X  t'°. 
Whence  V"  \  s°  =  s  :  r  —  v.      That  is  in  ali  Curves,  as 


FlG.   Il 


ATTEMPTS  AT  ARITHMETISATION    229 

the  Ultimator  of  the  Abscissa  is  to  the  Ultimator  of 
the  Semiordinate  ;  so  is  the  Semiordinate  itself  to 
the  Subnormal  "  (p.  51). 

The  author  has  occasion  to  use  second  and  third 
ultimators  afid  to  consider  ultimators  as  variable  or 
invariable.      He  lets  (p.  60)  i'°  be  the  invariable  of 

the  first  ultimator  ;ir°,  x  the  invariable  of  the  second 

ultimator  x^  etc. ,  and  warns  the  reader  that  his 
dot  does  not  mean  a  fluxion.  In  the  more  involved 
applications  to  curves  he  lets  an  infinitely  small  arch 
equal  ;ir°  =  i-°.  Our  impression  of  the  book  is  that 
the  author's  intentions  were  good  when  he  attempted 
an  arithmetisation.  But  there  is  a  total  lack  of  clear 
and  rigorous  exposition. 

200.  The  Ladies'  Diary^  London,  1750,  p.  45, 
contains  a  hostile  criticism  of  the  Doctrine  of 
Ultimators  by  an  anonymous  writer  (probably  the 
editor,  Robert  Heath),  in  which  the  author  of  this 
doctrine  is  said  to  declare  that  fluxions,  as  explained 
by  Newton,  are  "  absurd  and  unintelligible,"  and  to 
place  confidence  in  '*  the  Authority  of  a  certain  Irish 
B — p,  a  Mathematician  as  wise  as  himself.  For 
you  must  know  that  this  pious  B — p  (the  sagacious 
Author  of  the  Analyst,  as  he  stiles  him)  out  of  his 
religious  Zeal  against  Mathematical  Learning,  had 
been  engaged  in  the  same  senseless  Attempt  with 
himself,  of  degrading  the  noblest  Science.  .  .  . 
Having  thus,  as  he  [the  author  of  ''Ultimators"] 
thinks,  overturn'd  the  Doctrine  of  Fluxions  .  .  . 
he  has  given  us  instead  of  it  .    .   .   a  new  Science  of 


230  LIMirS    AND   FLUXIONS 

his  ovvn,  whose  Foundation,  it  seems,  depends  on 
Cyphers^  and  Nought  Powers  full  of  conceited  Ex- 
pressions.  .  .  .  He  expresses  his  Ultimators  by 
the  Help  of  ar°,  j°,  ^s°,  etc. ,  calls  them  peculiar  Units, 
and  of  different  Values,  ali  of  which  is  absurd.  .  .  . 
I  pass  over  ...  his  using  ;r**,  jj/°,  <c^,  for  the  same 
End  as  others  use  i-,  y,  è.  " 

John  Petvin^    I750 

20 1 .  Ina  Sketch  of  Universal  Arithmetic^ ^  brought 
out  as  a  posthumous  booklet,we  encounter  acuriosity. 
Its  philosophy  of  mathematics  and  of  fluxions  in 
particular  is  set  off  by  the  following  quotations  : 

(Page  156)  **I  do  not  then  consider  it  [mathe- 
matica! quantity]  as  generated  or  produced,  but  as 
that  which  is.  Time  and  Motion  produce  nothing 
of  the  Kind,  and  bave  no  Place  bere.  Nor  do  I 
consider  it  as  continuous,  nor  as  consisting  of  very 
small  or  infinitely  little  Parts,  but  as  consisting  of 
Parts  in  general.  These  Parts  therefore  I  con- 
sider as  discrete:  And  by  x^  y,  ^,  etc,  I  under- 
stand  Multitude.  The  Ones  or  Monads,  of  which 
X  is  many,  I  cali  x  \  .  ,  .  Nor  do  I  consider  ;r,  y,  z, 
etc. ,  barely  as  many  ;  but  as  a  certain  many.  So 
that  X,  y,  ,:■,  etc,  are  Wholes  ;  x,  y,  f,  etc,  their 
respective  Parts.  These  Parts  may  be  considered 
again    as   Wholes,   consisting  of  another    Order  of 

^  Letters  concerning  Mind.  To  which  is  added,  a  Sketch  of  Uni- 
versal Arithmetic  ;  comp7-ehending  the  Differential  Calculus,  aììd  the 
Doctrine  of  l'iuxions.  Hy  the  late  Reverend  Mr.  John  Petvin,  A.M., 
Vicar  of  Ilsington  in  Devon,  London,  1750. 


ATTEMPTS   AT  ARITHMETISATION    231 

Parts,"  designateci  as  x,  etc.  '*  Such  Things  as  an 
Instant,  a  Point,  a  Fluxion,  she  [arithmetic]  has 
nothing  to  do  with.  ...  1  have  joined  Fluxion  with 
Point  and  Instant,  because  Fluxion  seems  to  be  to 
Motion,  as  an  Instant  is  to  Time  ;  which  I  suppose 
to  be  as  a  Point  is  to  a  Line.  Motion  cannot  be 
conceived  without  Time  and  Space  ;  and  when  the 
former  runs  into  an  Instant,  and  the  latter  into  a 
Point,  then  it  is  (as  I  understand  it)  that  Motion 
becomes  Fluxion.  ...  In  this  Sense  Fluxion  is 
no  more  a  Part  of  Motion  than  a  Point  is  a  Part  of 
a  Line."  His  "  parts  "  are  finite  increments.  The 
part  oi  xyìsxy  -\-yx -\-xy.  *  '  This  Doctrine  of  Wholes 
and  Parts  proceeds  upwards  from  Parts  to  Wholes, 
as  well  as  downwards  from  Wholes  to  Parts  uni- 
versally  "  (p.  159).  "The  Ordinate  therefore  being 
x'"y  when  x"'x  expresses  the  Fluxion  of  it,  the  only 
Meaning  I  have  for  x  is,  that  it  is  the  Proportion  of 
a  Point  to  an  Instant.  And  to  my  Apprehension, 
a  Point  may  as  well  be  called  a  last  Line,  as  this 
called  a  Velocity."  ''I  have  lately  deduced  some 
arithmetical  Theorems  from  arithmetical  Principles, 
which  other  Mathematicians  have  drawn  from 
Fluxions  of  PTuxions,  etc. ,  and  these  Theorems  fell 
in  with  my  Design."  Just  how  these  deductions 
were  made  is  not  explained  by  the  author. 

[ohn  LandeUy  1758 

202.  John  Landen  was  a  self-educated  mathe- 
matician  of  real  mathematical  power.  Had  he  had 
the  benefits  of  University  training  he  might  have 


232  LIMITS   AND   FLUXIONS 

occupied  a  mudi  higher  rank  as  a  mathematician. 
Foreigners  place  him  high  among  his  English  con- 
temporaries.  He  wrote  M atheìnatical  Lucubrations^ 
1755,  and  Residuai  Analysis^  1764.  We  shall  con- 
sider  only  his  Discoui'se  concerning Residuai  Analysis^'^ 
1758.      From  it  we  quote  as  follows  : 

*' Yet,  notwithstanding  the  method  of  fluxions  is 
so  greatly  applauded,  I  am  induced  to  think,  it  is 
not  the  most  naturai  method.  .  .  .  The  operations 
therein  being  chiefly  performed  with  algebraic 
quantities,  it  is,  in  fact,  a  branch  of  the  algebraic 
art,  or  an  improvement  thereof,  made  by  the  help 
of  some  peculiar  principles  borrowed  from  the 
doctrine  of  motion.  .  .  .  We  may  indeed  very 
naturally  conceive  a  line  to  be  generated  by  motion  ; 
but  there  are  quantities  .  .  .  which  we  cannot 
conceive  to  be  so  generated.  It  is  only  in  a 
figurative  sense,  that  an  algebraic  quantity  can  be 
said  to  increase  or  decrease  with  some  velocity. 
Fluxions  therefore  are  not  immediately  applicable 
to  algebraic  quantities.  .  .  .  It  therefore,  to  me, 
seems  more  proper,  in  the  investigation  of  proposi- 
tions  by  algebra,  to  proceed  upon  the  aftciently- 
received  principles  of  that  art.  .  .  .  That  the  borrow- 
ing  principles  from  the  doctrine  of  motion,  with  a 
view  to  improve  the  analytic  art,  was  done,  not 
only  without  any  necessity,  but  even  without  any 
peculiar  advantage,  will  appear  by  showing,  that 
whatever  can  be  done  by  the  method  of  computa- 

^  A  Discourse  Concentiug  Residuai  Analysis  :  A  new  Branch  of  the 
Algebraic  Art.     By  John  Landen.     London,  1758. 


ATTEMPTS   AT  ARITHMETISATION    233 

tion,  which  is  founded  on  those  borrowed  prin- 
ciples,  may  be  done  as  well,  by  another  method 
founded  entirely  on  the  anciently-received  principles 
of  algebra.  .  .  .  It  is  by  means  of  the  foUowing 
theorem  [p.   5],  viz. 


in  nt 

Xn  —Vn  ^JI^i  X       Xj  X 

=  ;l'«      X 

x  —  v 


v\'^     v\^^     v\^^ 

(where  m  and  n  are  any  integers)  that  we  are 
enabled  to  perform  ali  the  principal  operations  in 
our  said  Analysis." 

His  Residuai  Analysis  is  a  method  involving 
vanishing  fractions  and  therefore  not  free  from  con- 
troversial  questions.  That  the  fluxion  of  x^  is  ^x'^ 
is  explained  according  to  the  Residuai  Analysis  by 
the  consideration  that  {j^  —  x^)  —  {}>  —x)=x'^^xy-\-y^, 
which  is  equal  to  3;^^  when  j;/  =  ;r.  We  proceed  to 
give  an  application  in  Landen's  own  words  : 

203.   (Page   5)   "  Fluxionists,   in  determining  the 

limit  of  the  ratio  of  the  increments  of  x  and  x"" , 
commonly  have  recourse  to  the  binomial  theorem 
(which  is  mudi  more  diffìcult  to  investigate  than  the 
limit  they  are  seeking)  :  But  how  easily  may  that 
limit  be  found,  without  the  help  of  that  theorem,  by 
the  equation  exhibited  in  page  5  !  Thus,  the  incre- 
ment  of  x  being  denoted   by  x\   the   increment  of 


x"*  is   x-{-x'»  —x'\    and    the    ratio    of   those    incre- 
ments is 


234 


LIMITS    AND   FLUXIONS 


X-\-X  *^  —X*^        X  +  X  **  —X' 


x+x'  —X 


=x-\-x' 


-1 


1+  ^ 

X 

1 

'           X 

^x+x' 

3 

(?«) 

V 

'  x+x'  '  XJtx' 

,+  ^ 

^            X    " 

?f^            X 

3^» 

r 

(«) 

'  x+x' 

x-^x' 

x-\-x 

which,   when   x    vanishes,    is    manìfestly    equal    to 

7/^    ^f  - 1 

— ;r«      ,  the  limit  of  the  said  ratio." 
n 

The  explanation  of  the  method  of  drawing  tan- 
gents  is  too  long  for  quotation,  and  we  shall  limit 
ourselves  to  the  following  outline  of  it,  as  given 
by  Landen  : 

*'  I  consider  the  curve  as  already  described,  with- 
out  any  regard  to  its  generation,  and  find  the  value 
of  a  certain  line  (terminated  by  the  curve  and 
its  tangent),  in  algebraic  terms  involving  {s)  the 
subtangent  with  other  quantities  ;  which  algebraic 
expression  I  observe,  from  an  obvious  property  of 
the  line  it  is  found  to  denote,  must  have  a  certain 
property  with  respect  to  being  positive  or  negative 
in  certain  cases.  I  therefore  assume  that  expres- 
sion equal  to  another  which  is  known  to  have  that 
very  property  ;  and  from  thence,  by  means  of  the 
theorem  mentioned  in  page  5,  readily  find  the  re- 
quired  value  o{  s"  (p.   io). 

204.  Landen's  Discourse  was  attacked  by.  an 
anonymous  writer  in  the  Monthfy  Review  for  June 
1759,  who  claims  that  the  Residuai  Analysis  "  is  no 


ATTEMPTS   AT  ARITHMETISATION    235 

other  than  Sir  Isaac  Newton's  method  of  differences  ; 
and  it  is  well  known,  that  if  the  differences  are 
diminished  so  as  to  vanish,  their  vanishing  ratio 
becomes  that  of  fluxions  "  (p.  560),  ''that  his  pre- 
tended  Residuai  Analysis  renders  the  investigations 
more  tedious  and  obscure  than  any  other.  "  Landen 
wrote  a  reply  in  the  July  number,  from  which 
we  quote  only  the  part  relating  to  the  word 
**function."  Says  Landen  :  ''He  objects  to  prime 
nuinber\  function,  etc,  as  terms  never  heard  before. — 
Alas  !  hovv  egregiously  does  he  betray  his  ignor- 
ance  !  " 

James  G lenze ^    1793 

205.  James  Glenie  graduated  from  the  University 
of  St.  Andrews,  and  became  a  military  engineer. 
He  was  a  prominent  Fellow  of  the  Royal  Society  of 
London.  In  his  Antecedental  Calculus,^  1793,  he 
begins  with  the  statement,  "  Having,  in  a  Paper, 
read  before  the  Royal  Society,  the  6th  of  March, 
1777,  and  pubhshed  in  the  Philosophical  Trans- 
actions  of  that  Year,  promised  to  deliver,  without 
any  consideration  of  Motion  or  Velocity,  a  Geo- 
metrica! Method  of  Reasoning  applicable  to  every 
purpose,  to  which  the  much  celebrated  Doctrine  of 
Fluxions  of  the  illustrious  Newton  has  been  or  can 

^  7'he  Antecedental  Calculus^  or  a  Geometrical  Method  of  Reasoning, 
without  any  Consideratiojt  of  Motion  or  Velocity  applicable  to  every 
Purpose^  to  which  Fluxions  have  been  or  can  be  applied.  By  James 
Glenie,  Esq.,  M.A.  and  F.R.S.  London,  1793.  According  to  G. 
Vivant!  (see  M.  Cantor's  Vorlesungen  uber  Geschichte  der  Mathematik, 
voi.  iv,  Leipzig,  1908,  p.  667),  James  Glenie  (1750-1817)  was  an 
artillery  officer  in  the  war  of  the  American  Revolution,  later  professor 
of  mathematics  in  the  military  school  of  the  East  India  Company. 


236  LIMTTS    AND   FLUXIONS 

be,  applied  ;  and  having  taken  notice  of  the  same 
Method,  in  a  small  Performance,  written  in  Latin, 
and  printed  the  i6th  of  July,  1776,  I  now  proceed 
to  fulfil  my  promise  with  as  much  conciseness  as 
perspicuity  and  precision  will  admit  of."  In  his 
A?itecedental  Calculus^  p.  io,  he  says  of  Newton  : 
*  I  am  perfectly  satisfied,  that  had  this  great  Man, 
discovered  the  possibility  of  investigating  a  general 
Geometrica!  Method  of  reasoning,  without  introduc- 
ing  the  ideas  of  Motion  and  Time,  ...  he  would 
have  greatly  preferred  it,  since  Time  and  Motion 
have  no  naturai  or  inseparable  connection  with 
pure  Mathematics.  The  fluxionary  and  differential 
Caculi  are  only  branches  of  general  arithmetical 
proportion." 

Glenie  speaks  (p.  3)  of  ''  the  excess  of  the  magni- 
tude,  which  has  to  B  a  ratio  having  to  the  ratio  of 
A  +  N  to  B  the  ratio  of  R  to  Q  (when  R  has  to  O 
any  given  ratio  whatever),  above  the  magnitude, 
which  has  to  B  a  ratio  having  to  the  ratio  of  A  to  B 
the  same  ratio  of  R  to  0,  is  geometrìcally  expressed 
by  "  a  complicated  fraction  vvhose  denominator  is 
B(R-Q)/Q^  and  whose  numerator  is  the  result  of  ex- 
panding  by  the  binomial  theorem  (A  -f  N)^^^  and 
then  subtracting  A^^^  therefrom. 

A  similar  expression  is  given  for  the  case  in  which 
A  — N  takes  the  place  of  A  +  N:  '*The  excess  of 
the  magnitude,  which  has  to  B  a  ratio,  having  to  the 
ratio  of  A  to  B  the  ratio  of  R  to  Q,  above  the  magni- 
tude, which  has  to  B  a  ratio,  having  to  the  ratio  of 
A  —  N  to  B  the  ratio  of  R  to  Q,   is  geometrically 


ATTEMPTS   AT  ARITHMETISATION    237 

expressed  by  "  a  fraction  whose  denominator  is 
g(R-Q)/Q^  and  whose  numerator  is  obtained  by  ex- 
panding  and  simplifying  Ai^/^-(A  -  N)R/Q.  *'But 
if  A  +  N  and  A  —  N  stand  to  B  in  relations  nearer 
to  that  of  equality  than  by  any  given  or  assigned 
magnitude  of  the  same  Kind,  these  general  expres- 
sions  become  R/0  .  A^^-QVQ.  N-fB^^-QVQ.  This  I 
cali  the  antecedental  of  the  magnitude  which  has 
to  B  such  a  ratio  as  has  to  the  ratio  of  A  to  B  the 
ratio  of  R  to  Q.      Now  if  N  the  antecedental  of  A 

be  denoted    by   À    or  A  .    .    .    [and]   if  Q  =  i    and 

"^AÀ 
R  =  2,    3,    4,    5,    etc. ,    this    expression  gives 


3A^A 
B* 


B 

respectively.  "      For  the  ''antecedent 


A  2AÀ 

A—    he    finds    the    '*  antecedental  "  or    2 A 

B  B 

2MÀ 
H-*^  T.      (putting  M  for  A  — B).      Glenie  shows  that 

at  a  point  of  a  curve  the  antecedentals  of  the  ab- 
scissa,  ordinate  and  curve,  are  as  the  sub-tangent, 
the  ordinate  and  the  tangent,  respectively. 

Glenie's  calculus  involves  extremely  complicated 
identities  of  ratios  and  examines  the  antecedents  of 
ratios  having  given  consequents.  The  style  of  ex- 
position  is  poor.  In  deriving  the  antecedentals, 
Glenie  quietly  drops  out  ali  the  terms  in  the 
numerator  that  involve  powers  of  N  higher  than 
the  first  power.  As  this  calculus  plays  no  part  in 
the  later  history  of  fluxions,  we  shall  give  only  one 
more  quotation  ;  it  relates  to  the  Binomial  Theorem 


238  LIMITS    AND   FLUXIONS 

(not  used  by  him  in  the  development  of  his  funda- 
mental  formulas).  He  says  (p.  ii)  :  '*  It  may  not 
perhaps  be  improper  to  add,  that,  if  to  the  ex- 
pressions  delivered  above  for  the  excess  of  the 
magnitude,  which  has  to  B  a  ratio,  having  to  the 
ratio  of  A  +  N  to  B,  the  ratio  of  R  to  O,  above  the 
magnitude,  which  has  to  B  a  ratio,  having  to  the 
Ratio  of  A  to  B  the  same  ratio  of  R  to  O  ;  and  for 
the  excess  of  the  magnitude,  which  has  to  B  a  ratio, 
having  to  the  ratio  of  A  to  B  the  ratio  of  R  to  Q, 
above  the  magnitude,  which  has  to  B  a  ratio,  having 
to  the  ratio  of  A  — N  to  B  the  ratio  of  R  to  O,  be 
prefixed  the  magnitude,  which  has  to  B  a  ratio, 
having  to  the  ratio  of  A  to  B  the  ratio  of  R  to  Q, 
we  get  a  geometrical  Binomial,  of  which,  when  it  is 
supposed  to  become  numerical,  the  famous  Binomial 
Theorem  of  Sir  Isaac  Newton  is  only  a  particular 
case.  " 

Reuiarks 

206.  The  classic  treatment  of  fluxions  in  Great 
Britain,  during  the  eighteenth  century,  rests  prim- 
arily  on  geometrical  and  mechanical  conceptions. 
Attempts  to  found  the  calculus  upon  more  purely 
arithmetical  and  algebraical  processes  are  described 
in  this  chapter.  Ali  these  attempts  are  either  a  com- 
plete failure  or  so  complicated  as  to  be  prohibitive. 
Easily  the  ablest  among  these  authors  was  John 
Landen.  De  Morgan  says  of  his  Analysis^:  "  It 
is  the  limit  of  D'Alembert  supposed  to  be  attaìned, 

^  Penny  CycloJ>iedia,  Art.  "  Differential  Calculus." 


ATTEMPTS  AT  ARITHMETISATION    239 

instead  of  beìng  a  terminus  which  can  be  attained  as 
near  as  we  please.  A  little  difference  of  algebraical 
suppositions  makes  a  fallacious  difference  of  form  : 
and  though  the  residuai  analysis  draws  less  upon 
the  disputable  part  of  algebra  than  the  method  of 
Lagrange,  the  sole  reason  of  this  is  that  the  former 
does  not  go  so  far  into  the  subject  as  the  latter." 

In  the  same  article  De  Morgan  speaks  of  Kirkby's 
Ultimators  thus  : 

'*  A  something  between  Landen  and  D'Alembert, 
as  to  principle,  published  in  1748,  was  called  the 
*  Doctrine  of  Ultimators,  containing  a  new  Acquisi- 
tion,  etc. ,  or  a  Discovery  of  the  true  and  genuine 
Foundation  of  what  has  hitherto  mistakenly  pre- 
vailed  under  the  improper  names  of  Fluxions  and 
the  Differential  Calculus.  '  " 


CHAPTER  X 
LATER   BOOKS  AND   ARTICLES   ON    FLUXIONS 

Encyclopcedia  Britannica^  ^77^,  1/79)  ^797 

207.  The  article  ''Fluxions"  in  the  first  edition 
of  the  Encyclopcedia  Britannica,  Edinburgh,  1771, 
gives  this  definition  :  "The  fluxion  of  any  magni- 
tude  at  any  point  is  the  increment  that  it  would 
receive  in  any  given  time,  supposing  it  to  increase 
uniformly  from  that  point  ;  and  as  the  measure  will 
be  the  same,  whatever  the  time  be,  we  are  at  liberty 
to  suppose  it  less  than  any  assigned  time."  The 
fluxion  of  a  rectangle  is  the  increment,  with  the  small 
rectangle  at  the  corner  omitted  ;  the  latter  *'  is  owing 
to  the  additional  velocity  wherewith  the  parallelo- 
gram  flows  during  that  time  and  therefore  is  no 
part  of  the  measure  of  the  fluxion."  "The  incre- 
ment a  quantity  receives  by  flowing  for  any  given 
time,  contains  measures  of  ali  the  different  orders  of 
fluxions;  for  if  it  increases  uniformly,  the  whole  in- 
crement is  the  first  fluxion  ;  and  it  has  no  second 
fluxion.  If  it  increases  with  a  motion  uniformly 
accelerated,  the  part  of  the  increment  occasioned  by 

the  first   motion  measures  the  first  fluxion,  and  the 

240 


LATER  BOOKS  AND  ARTICLES      241 

part   occasioned   by  the   acceleration   measures   the 
second  fluxion.    ..." 

The  same  article  is  reprinted  in  the  second  edition 
(1779)  and  the  third  edition  (1797). 

Robert  Thorp,  lyyy 

208.  Thorp  made  a  translation  of  part  of  New- 
ton's  Principia.'^ 

In  the  **  advertisement  "  we  read  :  '*  The  doctrine 
of  prime  and  ultimate  ratios  .  .  .  is  established,  so 
as  to  remove  the  various  objections  which  have  been 
raised  against  it,  since  it  was  first  published.  To 
the  relations  of  finite  quantities  alone  the  reasoning 
on  this  subject  is  confined."  The  translation  of 
quantitates  quam  mininice,  evanescentes^  ultimce^  in- 
finite rnagnce^  and  the  like,  has  not  been  literal,  yet 
they  are  "  explained  in  that  sense  under  which  the 
author  cautions  his  readers  to  understand  them. 
This  is  the  more  necessary,  as  the  terms  infinite^ 
infinitesimal,  least  possible^  and  the  like,  have  been 
grossly  misapplied  and  abused." 

209.  In  the  Commentary  to  Lemma  i  in  Sect.  i 
of  Bk.  I  in  the  Principia,  Thorp  says  :  *  *  The  prime 
and  ultimate  ratios  of  magnitudes  .  .  .  are  investi- 
gated  by  observing  their  finite  increments  or  decre- 
ments,  and  thence  finding  the  limits  of  the  ratios 
of  those  vai'iable  magnitudes  ;  not  the  ratios  to 
which    the    magnitudes    ever    actually    arrive    (for 

^  Mathematical  Principks  of  Naturai  Philosophy.  By  Sir  Isaac 
Newton,  Knight.  Translated  into  English,  and  illiistrated  with  a 
Commentary,  by  Robert  Thorp,  M.A.,  voi.  i,  London,  1777. 

16 


242  LIMITS   AND  FLUXIONS 

they  are  never,  strictly  speaking,  either  prime  or 
ultimate  in  fact),  but  those  limits  to  which  the 
ratios  •  of  magnitudes  perpetually  approach  ;  which 
they  can  never  reach,  nor  pass  beyond  ;  but  to 
which  they  appear  nearer  than  by  any  assignable 
difìference.  "  .  .  .  "  We  now  proceed  to  explain  this 
Lemma  more  particularly  than  perhaps  might  seem 
necessary,  had  it  not  been  much  controverted,  mis- 
represented,  and  misunderstood.  "  As  one  of  the 
conditions  of  the  proposition,  Thorp  states,  is  "  that 
quantities  and  the  ratios  of  quantities  must  con- 
tinually  tend  to  equality.  The  one  must  never 
become  equal  to,  nor  pass  beyond  the  other  :  their 
difference  must  never  either  vanish  to  nothing,  or 
become  negative."  In  this  restriction  Thorp  goes 
even  further  than  had  Robins.  The  following 
passage  from  Thorp's  commentary  is  thoroughly  in 
the  spirit  of  Robins  :**...  That  we  may  not  be  led, 
from  the  expression  ultiniately  equal,  to  suppose, 
that  there  is  an  ultimate  state,  in  which  they  are 
actually  equal,  we  are  cautioned  in  the  scholium  at 
the  end  of  this  Section  [of  Principia^  Bk.  I,  Sect.  i] 
in  these  words,  The  ultimate  ratioSy  in  which  quantities 
vanishy  are  not  in  reality  the  ratios  of  ultimate 
quantities  ;  but  the  limits  to  which  the  ratios  of 
quantities  continually  decreasing  always  approach  ; 
which  they  never  can  pass  beyond ^  nor  arrive  at^  unless 
the  quantities  are  continually  and  indefinitely  dimin- 
ished,  According  to  Thorp,  the  inscribed  or  cir- 
cumscribed  polygon  can  never  arrive  at  the  curve. 
He    quotes    from    Saunderson's   Fluxions.      By   the 


LATER  BOOKS  AND  ARTICLES      343 

doctrine  of  indivisibles  there  *'has  been  introduced 
into  mathematica!  reasoning  ali  that  absurd  jargon 
concerning  quantities  infinitely  great,  and  infinitely 
little,  which  has  been  so  much  objected  to  by  mathe- 
maticians.  And,  though  it  has  often  been  elegantly 
applied  by  some  able  geometers  to  the  demonstra- 
tion  of  many  noble  theorems  ;  yet  in  the  hands  of 
less  accurate  reasoners,  it  has  often  led  to  false 
conclusions"  (p.   71). 

F.   Holliday,    1777 

210.  In  a  somewhat  lengthy  preface  to  his  Intro- 
duction  to  Fluxions^  the  author  tells  that,  when  in 
1745  he  was  in  London,  in  company  with  W.  Jones 
and  De  Moivre,  they  expressed  great  approbation 
of  Emerson's  Fluxions^  with  regard  to  the  method 
of  treatment,  but  thought  his  book  too  high  for 
beginners.  The  author  tries  to  be  more  diffuse  in 
the  laying  down  of  first  principles.  He  derives  the 
fundamental  results  in  two  ways  :  first,  by  the  aid 
of  nascent  or  evanescent  quantities,  as  suggested  by 
Newton's  Principia;  second,  ''without  using  any 
infinitely  small  quantities,  or  vanescent  Parallelo- 
grams,  which  perhaps  will  be  more  acceptable  to 
many  of  my  Readers.  "  Holliday  explains  at  great 
length  the  Scholium  (see  our  §§  10-15)  on  prime 
and  ultimate  ratios,  and  gives  a  short  account  of 
the  invention  of  fluxions  as  given  in  the  review  of 

^  An  hitroduction  to  Fluxions^  Designedfor  the  Use,  and  Adapted  to 
the  Capacities  of  Beginners.  By  the  Reverend  F.  Holliday,  Vicar  of 
West  Markham  and  Bothamsall,  Nott's.     London,  1777. 


244  LIMITS   AND  FLUXIONS 

CoUins's  Commercium  Epistolicurn  in  the  Philosophical 
TransactionSy  17 17.  Though  following  Newton 
closely,  variations  were  bound  to  arise.  Thus, 
Holliday  says  (p.  73),  *'  Fluxions  are  not  magnitudes 
but  the  velocities  with  which  magnitudes,  varying  by 
a  continuai  motion,  increase  or  decrease."  It  cannot 
be  claimed  that  HoUiday  made  any  contribution  to 
the  philosophy  of  fluxions,  nor  even  that  he  profited 
as  much  as  he  might  by  the  refinements  in  the  logie 
which  had  been  made  by  English  writers  since  the 
time  of  Newton. 

Charles  Hutton,   1796,   1798 

211.  In  his  M athematical  Dictionary^  London, 
1796,  Charles  Hutton  makes  reference  to  the 
advantage  of  Simpson's  definition  of  a  fluxion  as  a 
magnitude  uniformly  generated  in  a  finite  time,  the 
imagination  being  now  no  longer  confined  to  a  single 
point  and  to  the  velocity  at  that  point  ;  moreover, 
*  '  higher  orders  of  Fluxions  are  rendered  much  more 
easy  and  inteUigible.  " 

212.  From  the  part  on  fluxions  in  Hutton's  Course 
of  Mathematics  ^  we  take  the  following  : 

*  *  The  rate  or  proportion  according  to  which 
any  flowing  quantity  increases,  at  any  position  or 
instant,  is  the  Fluxion  of  the  said  quantity,  at 
that  position  or  instant  :  and  it  is  proportional  to 
the  magnitude  by  which  the  flowing  quantity  would 
be  uniformly  increased,   in   a  given   time  with   the 

^  A  Coìirseof  Mathematics.     By  Charles  Hutton.     London,  4th  ed., 
1803-1804,  voi.  ii,  p.  279.     [First  ed.,  1798-1801.] 


LATER   BOOKS   AND   ARTICLES      245 

generating  celerity  uniformly  contìnued  during  that 
time." 

*'.  .  .  If  the  motion  of  increase  be  accelerated, 
the  increment  so  generated,  in  a  given  finite  time, 
will  exeeed  the  fluxion  :  .  .  .  But  if  the  time  be 
indefinitely  small,  so  that  the  motion  be  considered 
as  uniform  for  that  instant  ;  then  these  nascent 
increments  will  always  be  proportional,  or  equal,  to 
the  fluxions,  and  may  be  substituted  instead  of  them 
in  any  calculation." 

The  fluxion  oi  xy  is  derived  in  two  ways  :  the  first 
by  the  method  of  considering  the  rectangle  composed 
of  two  parts,  as  previously  expounded  by  Rowe. 

The  second  method  finds  algebraically  the  incre- 
ment xy  +yx'' +xy,  "  of  which  the  last  term  x'y  is 
nothing,  or  indefinitely  small,  in  respect  of  the  other 
two  terms,  because  x'  and  y  are  indefinitely  small 
in  respect  of  x  and  y.  .  .  .  Hence,  by  substitut- 
ing  X  and  j>  for  x'  and  y,  to  which  they  are  propor- 
tional, there  arises  xy-\-yx  for  the  true  value  of  the 
fluxion  oi  xy." 

S.    Vince,  1795,  1805 

213.  Vince's  Principles  of  Fluxions  appeared  in 
1795    as    the    second    volume    of  the    Principles  of 

Mathematics  and  Naturai  Philosophy  in  Four 
VolumeSy^    which     were     brought     out     under    the 

general    editorship    of    James    Wood.       A    second 

^  The  Principles  of  Mathematics  and  Naturai  Philosophy  in  Four 
Volumes.  Voi.  II,  The  Principles  of  Fluxions  :  Designed  for  the  Use 
of  Students  in  the  University.  By  the  Rev.  S.  Vince,  A.ÀI.,  F.K.S., 
Cambridge,  1795, 


246  LIMITS    AND   FLUXTONS 

edition  of  Vince  vvas  printed  in   1805.      From  thìs 
second  edition  we  quote  : 

P.  I  :  *'The  velocities  with  which  flowing  quan- 
tities  increase  or  decrease  at  any  point  of  time,  are 
called  ih.e.  Jluxìons  of  those  quantities  at  that  instant. 

"  As  the  velocities  are  in  proportion  to  the 
increments  or  decrements  uniformly  generated  in 
a  given  time,  such  increments  or  decrements  will 
represent  the  fluxions.  "^ 

Vince  also  quotes  Newton  on  the  generation  of 
quantities  by  motion  :  '*Sir  I.  Newton,  in  the 
Introduction  to  his  Quadrature  of  Curves^  observes 
that  '  these  geneses  really  take  place  in  the  nature 
of  things,  and  are  daily  seen  in  the  motion  of  bodies. 
And  after  this  manner,  the  ancients,  by  drawing 
moveable  right  lines  along  immoveable  right  lines, 
taught  the  geneses  of  rectangles.'" 

Vince  gives  no  formai  definition  of  a  //;;///  ;  but 
his  philosophy  of  this  subject  is  disclosed  by  the 
two  following  quotations  (pp.  4  and  5):  "  By 
keeping  the  ratio  of  the  vanishing  quantities  thus 
expressed  by  finite  quantities,  it  removes  the 
obscurity  which  may  arise  when  we  consider  the 
quantities  themselves  ;  this  is  agreeable  to  the 
reasoning  of  Sir  I.  Newton  in  his  Principia^  Lib.  I, 
Sect.   I,  Lem.  7,  8,  9." 

"  It  has  been  said,  that  when  the  increments  are 

^  "This  is  agreeable  to  Sir  I.  Newton's  ideas  on  the  subject.  He 
says  :  '  I  sought  a  method  of  determining  quantities  from  the  velocities 
of  the  motions  or  increments  with  which  they  are  generated  ;  and  call- 
ing  these  velocities  of  the  motions  or  increments,  fluxions^  and  the 
generated  quantities  fluents,  I  fell  by  degrees  upon  the  method  of 
fluxions.'— Introd.  to  Quad.  Curves" 


LATER   BOOKS  AND  ARTICLES     247 

actually  vanished,  it  is  absurd  to  talk  of  any  ratio 
between  them.  It  is  true  ;  but  we  speak  not  here 
of  any  ratio  then  existing  between  the  quantities, 
but  of  that  ratio  to  which  they  have  approached  as 
their  liniit  ;  and  that  ratio  stili  remains.  Thus,  let 
the  increments  of  two  quantities  be  denoted  by 
ax^  +  inx  and  bx^  +  nx  ;  then  the  limit  of  their  ratio, 
when  ;r  =  o,  is  m  :  n  \  for  in  every  state  of  these 
quantities,  ax'^-\-mx  :  bx^-{-nx  :  :  ax-{-7n  :  òx  +  n  :  : 
(when  x=o)  m  :  n.  As  the  quantities  therefore 
approach  to  nothing,  the  ratio  approaches  to  that 
oi  m  \  n  as  it's  limit.  We  must  therefore  be  careful 
to  distinguish  between  the  ratio  of  two  evanescent 
quantities,  and  the  limit  of  their  ratio  ;  the  former 
ratio  never  arriving  at  the  latter,  as  the  quantities 
vanish  at  the  instant  that  such  a  circumstance  is 
about  to  take  place." 

By  aid  of  the  binomial  theorem,  Vince  finds  the 
fluxion  of  ;ir",  when  the  fluxion  of  ;ir  is  given  ;  he  then 
finds  the  fluxion  of  xy  by  considering  {x •\- yf  =  x'^ -\- 
2xy+y'^y  by  which  the  fluxion  of  2xy  can  be  found 
in  terms  of  the  fluxions  (x-^yY,  x^  aindy^. 

Agnesi — Colson — Hellins^  1801 

214.  The  Analytical  Institutions^  is  the  first  cal- 
culus  that  was  written  by  a  woman.      The  authoress 

^  Analytical  Institutions ,  in  four  books  :  Originally  written  in 
Italian,  by  Donna  Maria  Catana  Agnesi,  Professor  of  the  Mathe- 
maticks  and  Philosophy  in  the  University  of  Bologna.  Translated 
into  English  by  the  late  Rev.  fohn  Colson,  A/.  A.,  F.P.S.,  and  Lticasian 
Professor  of  the  Mathematicks  in  the  University  of  Cambridge.  Now 
first  printed,from  th»  Translators  Manuscript ,  under  the  inspection  of 
the  Rev.fohn  Hellins,  B.D.,  F.R.S.     Vols.  i  and  il.     London,  rSoi. 


248  LIMITS    AND   FLUXIONS 

is  the  noted  Maria  Gaetana  Agnesi,  of  the  University 
of  Bologna. 

The  Italian  originai  was  first  published  at  Milan 
in  1748.  The  two  volumes  of  the  translation  were 
printed  at  the  expense  of  Baron  Maseres.  In  an 
introduction,  Hellins  points  out  that  Colson  hoped 
to  interest  the  ladies  of  England  in  the  study  of 
fluxions  by  his  translation  of  the  work  of  the  great 
Italian  lady,  "  And,  in  order  to  render  it  more  easy 
and  useful  to  the  Ladies  of  this  country,  ...  he 
[Colson]  had  designed  and  begun  a  popular  account 
of  this  work,  under  the  title  of  The  Pian  of  the 
Lady's  System  of  Analyticks  ;  explaining,  article  by 
article,  what  was  contained  in  it.  But  this  he  did 
not  live  long  enough  to  finish." 

215.  Colson  dealt  wìth  Agnesi's  work  somewhat 
as  Stone  had  dealt  with  that  of  De  L'Hospital, 
inasmuch  as  both  translators  substituted  the  nota- 
tion  of  Newton  in  place  of  that  of  Leibniz.  The 
word  fluxions  (''flussioni")  occurs  in  the  originai 
Italian  of  Agnesi's  masterly  work.  How  Colson's 
conscience  may  have  troubled  him,  when  a  fluxion 
stood  out  in  his  translation  as  something  "  infinitely 
little,"  may  be  judged  when  we  consider  that  in 
1736  he  brought  out  an  English  translation,  with 
an  extensive  comment,  of  Newton's  Method  of 
Fluxions.  With  Newton  a  fluxion  always  meant 
a  velocity. 

We  quote  a  few  passages  from  Colson's  Agnesi 
(voi.  ii,  pp.   1,2): 

"The    Analysis    of  infinitely    small    Quantities, 


LA  TER   BOOKS  AND  ARTICLES     249 

which  is  otherwise  called  the  Differential  Calculus^ 
or  the  Method  of  Fluxions^  is  that  which  is  con- 
versant  about  the  differences  of  variable  quantities, 
of  whatever  order  those  differences  may  be.  " 

"Any  infiiiitely  little  portion  of  a  variable  quantity 
is  called  it's  Difference  or  Fluxion  ;  when  it  is  so 
small,  as  that  it  has  to  the  variable  itself  a  less  pro- 
portion  than  any  that  can  be  assigned  ;  and  by 
which  the  same  variable  being  either  increased  or 
diminished,  it  may  stili  be  conceived  the  same  as 
at  first." 

On  p.  3  we  read  that  certain  lines  in  a  figure 
"  will  be  quantities  less  than  any  that  can  be  given, 
and  therefore  will  be  inassignable ^  or  differentials^  or 
infinitesimals^  or,  finally,  fluxions.  Thus,  by  the 
common  Geometry  alone,  we  are  assured  that  not 
only  these  infinitely  little  quantities,  but  infinite 
others  of  inferior  orders,  really  enter  the  composi- 
tion  of  geometrical  extension." 

"These  propositions,"  says  a  reviewer^  of  the 
translation,  '*may  appear  exceptionable,  in  point  of 
language,  to  the  rigorists  in  geometry  ;  but  they  are 
nevertheless  founded  on  good  principles,  and  furnish 
rules  for  the  comparison  of  evanescent  quantities, 
which  will  prove  safe  guides  in  investigation.  The 
demonstrations  appear  to  us  to  be  perfectly  sound 
(if  the  word  infinite  be  taken  in  its  true  sense,  as 
denoting  merely  the  absence  of  any  limit),  with  the 
exception,  perhaps,  of  the  first  theorem,  which 
(as  is  not  a  little  curious  to  remark)  is  liable  to  the 

*    Edinburgh  Review^  voi.  iii,  1805,  p.  405, 


2  50  L1MTTS    AND   FLUXIONS 

same  objection  that  has  been  macie  of  ^the  lemma 
of  Newton's  Principia.  In  both  instances,  also,  the 
error  is  rather  apparent  than  real.  "  The  first  theorem 
in  question  states  that  the  two  intersecting  perpendì- 
culars  to  a  curve  drawn  at  the  ends  of  *'  an  infìnitely 
little  portion  of  it  of  the  first  order,"  *'may  be 
assumed  as  equal  to  each  other."  We  wonder  what 
Robins  and  Maclaurin  would  bave  thought,  had  they 
been  alive  in  i8oi  and  1805,  and  read  these  defini- 
tions  and  comments  !  What  horrible  visions  would 
these  ghosts  of  departed  quantities  bave  brought  to 
Bishop  Berkeley,  had  he  been  alive  !  But  the  nine- 
teenth  century  was  destined  to  bring  back  to  British 
soil  stili  greater  accentuations  of  infinitesimals. 

T.    Newton,    1805 

216.  The  Rev.  T.  Newton  says  in  the  preface  of 
bis  lllustrations  of  Sir  Isaac  Newton's  Method  :  ^ 

*  '  Every  Mathematician  now  considers  the  whole 
doctrine  of  Prime  and  Ultimate  Ratios  in  no  other 
light,  than  as  a  Doctrine  of  Limits.  "  Young 
readers  of  Sir  Isaac  Newton's  Principia  encounter 
difficulties  because  commentators  bave  made  ''use 
of  the  terms  of  Indivisibles,  in  their  explana- 
tions  ;  .  .  .  Newton  expressly  says,  that  by  the 
ultimate  ratios  of  quantities  he  means  the  ratios  of 
their   limits.  2      And   when    he   wants   to   infer    the 

^  An  Illitstration  of  Sir  Isaac  Newton^ s  Method  of  Reasonitì}(.  By 
Prime  and  Ultimate  Ratios.  By  the  Rev.  T.  Newton,  Kector  of  Tewin, 
Herts  ;  late  Fellow  of  Jesus  College,  Cambridge.     Leeds,  1805. 

'^  Seeour§§  12,  15. 


LATER   BOOKS   AND   ARTICLES      251 

equality  of  inequality  of  those  limits  from  some 
relation  of  the  variable  quantities,  which  are  never 
supposed  absolutely  to  reach  their  limits,  it  cer- 
tainly  requires  something  more  than  a  definition  to 
shew  this.  ...  It  is  not  my  intention  to  detain 
the  reader,  with  answering  the  objections  of  the 
Arialyst  and  his  foUowers,  because  it  has  been 
already  done  by  others  in  a  satisfactory  manner. 
.  .  .  Notwithstanding  the  assertions  of  some 
modem  writers,  the  method  of  ultimate  ratios  is 
extremely  perspicuous,  strictly  logicai,  and  more 
concise  than  any  other  of  modem  invention  ;  .  .  . 
it  neither  involves  the  strange  notion,  that  a 
straight  line  may  be  a  part  of  a  curve,  and  a  piane 
superficies  a  part  of  a  concave  or  convex  one  ;  nor 
the  unintelligible  idea  of  adding  and  subtracting 
indivisibles,  or  inconceivably  small  magnitudes. 
Whatever  magnitudes  are  compared,  according  to 
this  method,  they  are  always  supposed  to  be 
finite." 

T.  Newton  begins  with  the  following  two  defini- 
tions  (p.   i)  : 

'*If  a  variable  quantity,  either  increasing  or  de- 
creasing,  approaches  to  a  fixed  quantity,  the  differ- 
ence  between  them  being  continually  diminished,  so 
as  at  length  to  become  less  than  any  assignable 
quantity  ;  the  fixed  quantity  is  called  the  Limit  of 
the  variable  quantity." 

''  If  the  ratio  of  two  variable  quantities  continu- 
ally approaches  to  a  fixed  ratio,  so  as  to  come 
nearer  to  it  than  by  any  assignable  difference  ;  the 


252  LIMITS    AND   FLUXIONS 

fixed    ratio    is    called    the    Limiting    Ratio    of   the 
variable  quantities.  " 

William  Dealtry,    1810,    18 16 

217.  In  the  preface  of  Dealtry's  Principles  of 
Fluxions  ^  (  1 8 1 6)  we  read  : 

"The  method  of  Fluxions  rests  upon  a  principle 
purely  analytical  ;  namely,  the  theory  of  Hmiting 
ratios  ;  and  the  subject  may  therefore  be  considered 
as  one  of  pure  mathematics,  without  any  regard  to 
ideas  of  time  and  velocity.  But  the  usuai  manner 
of  treating  it  is  to  employ  eonsiderations  resulting 
from  the  theory  of  motion.  This  was  the  pian  of 
Sir  Isaac  Newton  in  first  delivering  the  principles  of 
the  method  ;  and  it  is  adopted  in  the  foUowing 
Work,  from  the  belief,  that  it  is  well  adapted  for 
illustration." 

Dealtry  defines  a  ''fluxion  of  a  quantity  at  any 
point  of  time"  as  **its  increment  or  decrement, 
taken  proportional  to  the  velocity  with  which  the 
quantity  flows  at  that  time."  .    .    . 

"  When  a  quantity  increases  with  a  velocity  which 
continually  varies,  the  quantity,  which  measures 
the  fluxion,  is  a  limit  between  the  preceding  and 
succeeding  increments,  and  is  ultimately  equal  to 
either  of  them.  "  He  explains  that  *'the  word 
ultimately  is  intended  to  denote  that  particular 
instant,   when   the  time  is  diminished  sine  limite^''' 

^  The  Principles  of  Fluxions  :  Designed for  the  Use  of  Students  in  the 
Universities.  By  William  Dealtry,  B.D.,  F.R.S.,  late  Fellow  of 
Trinity  College,  Cambridge.     2nd  ed.,  Cambridge,   1816. 


LATER  BOOKS  AND  ARTICLES     253 

and  quotes  Newton's  Scholium,  Sect.  i,  in  the  Prin- 
cipia. He  points  out,  also,  that  if  x  increases  uni- 
formly,  x'^  increases  with  accelerateci  velocity,  and 
the  part  of  the  increment  x'^  is  the  efifect  of  the 
acceleration,  and  therefore,  by  his  definition  of 
fluxion,  to  be  "  omitted  in  taking  the  fluxions  " 
(P.   8). 

New  Editions,    1 801- 1809 

218.  William  Davis,  who  was  a  bookseller  in 
London  and  editor  of  the  Companion  to  the  Gentle- 
man's  Diaryy  appears  also  as  the  editor  of  new 
editions  of  three  dififerent  texts  on  fluxions.  In 
1801  he  saw  through  the  press  the  second  edition 
of  Maclaurin's  Treatise  of  Fluxions  \  in  1805  the 
third  edition  of  Thomas  Simpson's  Doctrine  and 
Application  of  Fluxions.  In  1809  appeared  the  fourth 
edition  of  John  Rowe's  Doctrine  of  Fluxions^  revised 
**by  the  late  William  Davis." 

Remarks 

219.  Among  some  of  the  authors  of  this  period 
there  is  less  concern  than  among  writers  of  former 
years  about  the  attainment  of  the  rigour  of  the 
ancients.  Perhaps  the  effects  of  the  revival  of  the 
ideals  of  Euclid  and  Archimedes  which  followed  the 
publication  of  the  Analyst  were  gradually  subsiding. 
It  would  not  be  fair  to  this  age  to  judge  its  mathe- 
matical  status  altogether  by  the  authors  which  we 
bave  selected.      There  was  a  movement  under  way 


254  LIMITS    AND  FLUXIONS 

at  this  time  which  is  reflected  in  the  literature  that 
will  be  under  consideration  in  the  next  chapter. 

Both  before  the  time  of  Berkeley's  Analyst  and 
after  the  time  of  Maclaurin's  Fluxions  there  appeared 
in  Great  Britain  texts  which  superposed  British 
symbols  and  phraseology  upon  the  older  Continental 
concepts.  The  result  was  a  system,  destitute  of 
scientific  interest.  Newton's  notation  was  poor  and 
Leibniz's  philosophy  of  the  calculus  was  poor.  That 
result  represents  the  temporary  survival  of  the  least 
flt  of  both  systems.  The  more  recent  international 
course  of  events  has  been  in  a  diametrically  opposite 
direction,  namely,  not  to  superpose  Newtonian 
symbols  and  phraseology  upon  Leibnizian  concepts, 
but,  on  the  contrary,  to  superpose  the  Leibnizian 
notation  and  phraseology  upon  the  limit-concept,  as 
developed  by  Newton,  Jurin,  Robins,  Maclaurin, 
D'Alembert,  and  later  writers. 


CHAPTER  XI 

CRITICISMS  OF  FLUXIONS  BY  BRITISH  WRITERS 
UNDER  THE  INFLUENCE  OF  D'ALEMBERT, 
LAGRANGE,  AND  LACROIX 

Review  of  Lagrange's  '■^  Fonctions  analytiques^''  1799 

220.  Important  is  a  review  ^  of  Lagraiige's  Théorie 
des  fonctions  analytiques^  which,  as  is  well  known,  is 
an  attempt  to  deduce  the  principles  of  the  calculus, 
diverted  of  ali  reference  to  infinitely  small  or  evan- 
escent  quantities,  limits  or  fluxions,  and  reduced  to 
the  algebra  of  finite  quantities.  The  reviewer  gives 
a  general  criticism  of  the  methods  of  fluxions  and  the 
difìferential  calculus.  He  discusses  the  principle  of 
motion  :  ''  It  will  not  be  denied  that  this  principle  is 
introduced  purely  for  the  purpose  of  illustration,  .  .  . 
on  th^  ground  of  convenience.  .  .  .  The  mathe- 
matica! principle,  on  which  the  doctrine  of  fluxions 
depends,  is  a  definition  .  .  .  and  fluxions  were 
defined  to  be  velocities.  .  .  .  Novv  velocity  is 
nothing  real,  but  is  only  the  relation  between  the 
space  described  and  the  time  of  describing  it  ; — of 
which  relation  we  have  a  clear  idea  when  the  motion 
is  uniform."    The  reviewer  continues  :   *'  In  variable 

^  Monthly  Review,  London,  voi.  xxviii,  1799,  Appendix. 
255 


256  LIMITS    AND  FLUXIONS 

motion,  however,  we  inquire  what  velocity  is  ;  and 

here   it   is   defined   to  be  the  relation   between  the 

space  which  would  be   described  were   the   motion 

continued   uniform  from  any  point,   and   the  time. 

Stili  difficulties  remained  ;  this  definition  might  con- 

vey  to  the  mind  a  general   idea   of  the   nature  of 

velocity,  but  was  of  no  mathematical  use,  since  the 

space  which  would  be  described  could  not  be  immedi- 

ately   ascertained   and    determined.      Another  step 

was  therefore  to  be  made,  and  which  was  made  by 

establishing  this  proportion  ;  if  V  be  the  velocity, 

S  the  space,  which  would  be  described,  and  T  the 

time,    S'   the    space    really    described,    and    T'    the 

S 
corresponding  time;  then   V  =  ^  =  ultimate  rjatio  of 

S' 

,    when  S'  and   T'  are   indefinitely   diminished.  " 

Again  he  says  : 

*'  On  the  ground  of  perspicuity  and  evidence,  the 
understanding  is  not  much  assisted  by  being  directed 
to  consider  ali  quantity  as  generated  by  motion  ; 
.  .  .  when  such  quantities  as  weight,  density,  force, 
resistance,  etc. ,  become  the  object  of  inquiry  .  .  . 
then  the  true  end  of  the  figurative  mode  of  speech, 
illustration,  is  lost.  .  .  .  That  which  happened  to 
Aristotle  has  happened  to  Newton  ;  his  foUowers 
have  bowed  so  implicitly  to  his  authority,  that  they 
have  not  exercised  their  reason.  The  method  of 
fluxions  had  never  so  acute,  so  learned,  and  so 
judicious  a  defender  as  Maclaurin  : —  yet  who- 
ever  consults  it  .    .    .    finds  the  author  speaking  of 


CRiriCISMS  BY  BRITISH    WRITERS     257 

'  causes  and  effects,' of  'the  springs  and  principles 
of  things,'  and  proposing  to  deduce  the  'relation 
of  quantities  by  comparing  the  powers  which  are 
conceived  to  generate  them  '  ; — will  be  convinced 
that  this  could  only  happen  from  so  able  a  mathe- 
matician  having  failed  to  seize  the  right  principles." 
**  If  English  mathematicians  first  adopted  Newton's 
method  from  veneration  to  him,  .  .  .  they  have 
since  persevered  in  it  (we  may  almost  say)  against 
conviction.  "  The  reviewer  claims  that  the  criticisms 
of  D'Alembert,  Torelli,  and  Landen  have  shown  that 
the  use  of  motion  is  unnecessary  and  unreal.  We 
have  given  citations  from  Landen  in  an  earlier 
chapter  (see  our  §§  202,  203).  D'Alembert  is 
quoted  as  saying  fifty  years  previous  : 

**  Introduire  ici  le  mouvement,  c'est  y  introduire 
une  idée  étrangère,  et  qui  n'est  point  nécessaire  à  la 
démonstration  :  d'ailleurs  on  n'a  pas  d'idée  bien  nette 
de  ce  que  c'est  que  la  vitesse  d'un  corps  à  chaque 
instant,  lorsque  cette  vitesse  est  variable.  La  vitesse 
n'est  rien  de  réel  ;  .  .  .  c'est  le  rapport  de  l'espace 
au  tems,  lorsque  la  vitesse  est  uniforme  ;  .  .  . 
Mais  lorsque  le  mouvement  est  variable,  ce  n'est 
plus  le  rapport  de  l'espace  au  tems,  c'est  le  rapport 
de  la  différentielle  de  l'espace  à  celle  du  tems  ; 
rapport  dont  on  ne  peut  donner  d'idée  nette,  que 
par  celle  des  limites.  Ainsi,  il  faut  nécessairement 
en  revenir  à  cette  dernicre  idée,  pour  donner  une 
idée  nette  à^s  fiuxiofis,'''^ 

^  Art.    "Fluxion"    in  Encyclopédie^   oti   Didionnaire  raisonné  des 
Sciences^  etc,  t.  6,  Paris,  1756. 

17 


258  LIMITS    AND   FLUXIONS 

221.  The  reviewer  states  that  foreign  mathe- 
maticians  have  written  treatises  in  which  motion  is 
entirely  excluded,  "and  in  some  of  these  treatises, 
the  principles  of  the  doctrine  in  question  have  been 
laid  down  with  a  considerable  degree  of  evidence 
and  exactness."  The  Residuai  Analysis  of  Landen 
rests  on  *'a  process  purely  algebraical  :  but  the 
want  of  simplicity  .  .  .  is  a  very  great  objection  to 
it."  The  reviewer  is  of  the  opinion  that  Euler  and 
D'Alembert  give  "  the  most  clear  and  precise  notions 
of  the  principles  on  which  the  differential  calculus 
is  established.  "  He  refers  to  Euler's  Institutiones 
calculi  differentialis,  1755.  D'Alembert,  says  the  re- 
viewer, *'  observes  that  the  method  is  really  founded 
on  that  of  prime  and  ultimate  ratios,  or  of  limits, 
which  latter  method  is  only  an  algebraical  transla- 
tion  of  the  former  ;  that,  in  fact,  there  are  no  such 
things  as  inftnitely  small  quantities  ;  and  that,  when 
such  quantities  are  mentioned,  it  is  by  the  adoption 
of  a  concise  mode  of  speech  for  the  purpose  of 
simplifying  and  abridging  the  reasoning  ; — that  the 
true  object  of  consideration  is  the  limit  of  the  ratio 
of  the  finite  differences  of  quantities." 

The  reviewer  continues  :  ''The  explanations  given 
by  Euler  and  D'Alembert,  beyond  ali  doubt,  deserve 
much  consideration,  yet  their  method  of  consider- 
ing  the  doctrine  of  fluxions  is  not  completely  satis- 
factory,  but  is  objectionable  on  two  grounds  :  first, 
that  we  have  no  clear  and  precise  notion  of  the 
ratio  of  quantities,  when  those  quantities  are  in 
their   vanishing    state,    or    cease   to   be    quantities  ; 


CRITICISMS   BY  BRITISH   WRITERS    259 

secondly,  the  connection  and  naturai  order  of  the 
Sciences  are  interrupted,  if  we  give  a  di^tinct  and 
independent  origin  to  that  which  in  fact,  is  a  branch 
of  analysis  derived  from  the  same  common  stock, 
whence  ali  the  other  branches  are  deduced.  "  Then 
follows  a  sympathetic  account  of  the  foundations 
for  the  calculus  laid  by  Lagrange  in  his  Thcorie  des 
fonctions  analytiques^  ^798-  In  passing,  the  reviewer 
remarks  that  **  Emerson,  Stone,  Simpson,  Waring, 
etc. ,  have  published  treatises  on  fluxions  ;  in  none 
of  which,  however,  are  the  principles  clearly  laid 
down." 

Review  of  a  Memoir  of  Stockler,  1799 

222.  In  the  same  journal^  there  is  a  review  of 
a  memoir  on  fluxions  written  by  the  Portuguese 
mathematician,  Garcao  Stockler,  who  modifies  the 
explanation  of  fundamentals  by  the  introduction  of 
a  '' hypothetical  fluxion  "  (a  uniform  velocity  that 
generates  a  quantity  equal  to  the  real  increment 
generated  during  the  actually  variable  motion),  which 
is  always  contained  between  the  proper  fluxions  at 
the  first  and  second  instant  under  consideration. 
By  diminishing  the  interval  of  time,  the  hypothetical 
fluxion  approaches  the  true  fluxion  more  nearly  than 
by  any  assignable  quantity.  Here  also,  the  real 
object  of  consideration  is  the  liinit.  The  reviewer 
argues  that  the  fundamental  principles  are  not 
new,  and  that  the  objections  to  Newton's  fluxions 
apply  equally  to  those  of  Stockler.      In  a  reply  to 

^  Monthly  Review ^  voi.  xxviii,  London,  1799,  p.  571. 


200  LIMirS    AND   FLUXIONS 

the  Monthly  Review,  Stockler  denies  the  reviewer's 
allegation  that  he  [Stockler]  supposed  quantity  to 
be  generated  by  motion.  *'The  idea  of  motion, 
and  the  idea  of  velocity,  are  too  particular  to  be 
admitted  into  a  general  theory  of  fluent  quantities.  "  ^ 

Review  of  Lacroix's   '^  Calcul  dìfférentiel"    1800 

223.  A  review  of  S.  F.  La  Croix's  Traìté 
dii  calcul  différentiel'^  served  as  the  occasion  of 
further  comments  and  criticisms  of  fundamental 
concepts  : 

''  Who  would  direct  his  ridicule  against  the  refine- 
ments,  subtleties,  and  trifling  of  the  schoolmen,  if  he 
read  what  has  been  written  by  some  men  who  were 
presumed  to  be  the  greatest  masters  of  reason,  and 
whose  employment  and  peculiar  privilege  consisted 
in  deducing  truth  by  the  justest  inferences  from 
the  most  evident  principles  ?  The  history  of  the 
differential  calculus,  indeed,  shows  that  even  mathe- 
maticians  sometimes  bend  to  authority  and  a  name, 
are  influenced  by  other  motives  than  a  love  of  truth, 
and  occasionally  use  (like  other  men)  false  meta- 
physics  and  false  logie.  No  one  can  doubt  this,  who 
reads  the  controversial  writings  to  which  the  inven- 
tion  of  fluxions  gave  rise  :  he  will  there  find  most 
exquisite  reasonings  concerning  quantities  which 
survived  their  grave,  and,  when  they  ceased  to 
exist,  did  not  cease  to  operate  ;  concerning  an  in- 
finite  derivation   of  velocities, — and   a   progeny  of 

^  Monthly  Review,  voi.  xxxii,  p.  497. 

^  Monthly  Review^  voi.  xxxi,  London,  1800,  p.  493. 


CRI  TI  CTS  MS  BY  BRITISH    WRITERS     261 

infinitesimals  smaller  than  the  *  moonshine's  wat'ry 

beams,'  and  more  numerous  than 

'  Autumnal  leaves  that  strow  the  brooks 
In  Vallombrosa.'    (Milton,  Par.  Lost^  i,  302.) 

**The  contemporaries  and  partizans  of  Newton 
were  men  infinitely  inferior  to  him  in  genius,  but 
they  had  zeal,  and  were  resolved  to  defend  his 
opinions  and  judgments.  Hence  they  undertook  the 
vindication  of  fluxions,  according  to  the  principles 
and  method  of  its  author  ;  although  it  may  be  fairly 
inferred,  from  the  dififerent  explanations  given  of 
that  doctrine  by  Newton  in  dififerent  parts  of  his 
Works,  that  Newton  himself  was  not  perfectly  satis- 
fied  of  the  stabiHty  of  the  grounds  on  which  he 
had  established  it." 

The  reviewer  quotes  (p.  497)  from  Lacroix's 
preface  : 

**These  notions  [velocities,  motions],  although 
rigorous,  are  foreign  to  geometry,  and  their  appHca- 
tion  is  difficult.  .  .  .  Properly  speaking,  fluxions 
were  to  him  [Newton]  only  a  means  of  giving  a 
sensible  existence  to  the  quantities  on  which  he 
operated.  The  advantage  of  the  method  of  fluxions 
over  the  differential  calculus  in ,  point  of  meta- 
physics,  consists  in  this  ;  that,  fluxions  being  finite 
quantities,  their  moments  are  only  infinitely  small 
quantities  of  the  first  order,  and  their  fluxions  are 
finite  ;  by  these  means,  the  consideration  of  in- 
finitely small  quantities  of  superior  orders  is  avoided. 
...  I  can  only  mention  a  method  which  Landen 
gave  in   1758,  to    avoid  consideration  of  infinity  of 


262  LIMITS   AND   FLUXIONS 

motions,  or  of  fluxions,  since  it  rests  on  a  very 
elegant  algebraic  theorem  which  cannot  be  given  in 
a  work  of  this  nature.  The  freedom  with  which 
Landen  divests  himself  of  national  prejudice  stamps 
a  remarkable  character  on  his  work  ;  he  is  perhaps 
the  only  English  mathematician,  who  has  acknow- 
ledged  the  inconvenience  of  the  method  of 
fluxions."  .    .    . 

*'We  can  always  descend  from  the  function  to 
the  differential  coefiìcient  or  from  the  primitive 
function  to  the  derived  function  :  but,  generally 
speaking,  the  reverse  step  is  attended  with  the 
greatest  difficulty." 

**The  rivals  of  Newton  thought  and  invented 
for  themselves  ;  had  they  been  influenced  by  his 
authority,  and  devoted  their  talents  to  the  perfec- 
tion  of  synthesis,  science  must  have  been  con- 
siderably  retarded.  To  the  improvement  of  the 
algebraical  analysis,  is  to  be  attributed  the  amazing 
advances  of  physical  astronomy."^ 

Review  of  Carnofs   ^  ^  Réflexiofts ,''  i8oi 

224.  In  the  Montili)'  Review'^  (London)  for  1801 
there  is  a  short  and  unimportant  account  of  Lazare 
N.  M.  Carnot's  new  book,  Réflexions  sur  la  viéta- 
physique  du  calcul  infinitésininl^  1797-  Carnot 
explains  the  correctness  of  results  obtained  by  the 
infinitesimal  calculus  of  Leibniz  on  the  theory  of 
compensation  of  errors — a  theory  which  had  been 

^  Monthly  Review,  voi.  xxxii,  p.  491. 

2  Monthly  Review^  voi.  xxxiv,  1801,  p.  463. 


CRITICISMS  BY  BRITISH    WRITERS     263 

advanced  much  earlier  by  Berkeley  in  his  Analyst. 
Mr  Philip  E.  B.  Jourdain  has  found  clear  indica- 
tions  of  this  theory  in  Maclaurin's  Fluxions  and  in 
Lagrange's  Théorie  des  fonctions  analytiques.  The 
method  of  limits  is  explained  by  Carnet  in  the 
manner  of  D'Alembert.  ''Of  fluxions,  indeed," 
says  the  reviewer,  "  as  founded  on  the  strange  basis 
of  velocity,  there  is  no  account." 

Robert    Woodhouse^    1803 

225.  In  1803,  Robert  Woodhouse  published  his 
Principles  of  A  nalytical  Calculation,  ^  Woodhouse  had 
graduated  B.  A.  at  Caius  College,  Cambridge,  in 
1795,  as  senior  wrangler.  He  then  held  a  scholar- 
ship  and  a  fellowship  at  Caius  College,  devoting 
himself  to  mathematics.  He  has  the  distinction  of 
being  the  first  to  strongly  encourage  the  study  in 
England  of  the  mathematical  analysis  which  had 
been  created  on  the  Continent  by  Swiss  and  French 
mathematicians.  In  his  Principles  of  Analytical 
Calculation  he  discussed  the  methods  of  infinitesi- 
mais  and  limits,  and  Lagrange's  theory  of  function, 
pointing  out  the  merits  and  defects  of  each.  ''  By 
thus  exposing  the  unsoundness  of  some  of  the 
Continental  methods,  he  rendered  his  general  support 
of  the  system  far  more  weighty  than  if  he  had 
appeared  to  embrace  it  as  a  blind  partisan.  "^ 

226.  The  ideas  set  forth  in  this  book  are,  on  the 

^  The  Princif^les  of  Analytical  Calculation^  by  Robert  Woodhouse, 
A.M.,  F.R.S.     Caniliridge,  1803. 

"^  Art.  "  Woodhouse,  Robert,"  in  Sidney  Lee's  Dictionary  of  National 
Biography. 


264  LTMITS    AND   FLUXIONS 

whole,  in  such  dose  agreement  with  those  advanced 
in  the  preceding  reviews,  that  the  query  naturally 
arises,  whether  Woodhouse  is  not  the  author  of 
those  reviews.  We  have  reached  no  final  decision 
on  this  point. 

In  the  preface  Woodhouse  passes  in  review  the 
difìferent  methods  of  establishing  the  foundations  of 
the  calculus.  He  criticises  the  use  of  motion  in  the 
proof  of  the  binomial  and  other  related  theorems. 
''  It  required  no  great  sagacity  to  perceive,  that  a 
principle  of  motion,  introduced  to  regulate  processes 
purely  algebraical,  was  a  foreign  principle."  If  the 
binomial  theorem  and  related  theorems  for  the 
development  of  a  function  be  established  by  algebra, 
independently  of  motion,  then  *  *  from  the  second 
term  of  this  expansion,  the  fluxion  or  differential  of 
a  quantity  may  be  immediately  deduced,  and  in  a 
particular  application,  it  appears  to  represent  the 
velocity  of  a  body  in  a  motion.  The  fluxionists 
pursue  a  method  totally  the  reverse  ;  they  lay  down 
a  principle  of  motion  as  the  basis  of  their  calculus, 
thence  deduce  some  of  the  first  processes,  and 
establish  the  binomial  theorem,  by  which  it  is  said, 
the  extraction  of  roots  may  be  effected.  .  .  .  The 
project  of  extracting  the  square  and  cube  roots  of 
algebraical  quantities  by  a  principle  of  motion,  is 
surely  revolting  to  the  common  sense." 

**  Of  his  own  method.  Newton  left  no  satisfactory 
explanation  :  those  who  attempted  to  explain  it, 
according  to  what  they  thought  the  notions  of  its 
author,  and  .   .    .    by  reasoning  which  fairly  may  be 


CRITICISMS  BY  BRITISH    WRITERS     265 

called  tedious  and  prolix.  Of  the  commentators  on 
the  method  of  fluxions,  Maclaurin  is  to  be  esteemed 
most  acute  and  judicious,  but  his  Introduction 
exhibits  rather  the  exertions  of  a  great  genius 
struggling  with  difficulties,  than  a  clear  and  distinct 
account  of  the  subject  he  was  discussing."  To 
remove  this  prolixity,  it  was  proposed,  conformably 
to  the  notions  of  Newton,  to  cali  in  the  doctrine  of 
prime  and  ultimate  ratios  or  of  limits.  Euler  and 
D'Alembert,  on  the  other  hand,  rejected  motion,  but 
retained  limiting  ratios,  failing,  however,  in  supply- 
ing  a  satisfactory  explanation  therefor.  Wood- 
house  is  the  earliest  English  mathematician  who 
speaks  in  respectful  and  appreciative  terms  of 
the  services  to  mathematics  rendered  by  Bishop 
Berkeley.  In  fact,  Woodhouse  admits  as  valid 
some  of  Berkeley's  objections  which  had  been 
declared  invalid.  The  methods  of  treating  the 
calculus  **all  are  equally  Hable  to  the  objection  of 
Berkeley,  concerning  the  fallacia  suppositionis,  or 
the  shifting  of  the  hypotJiesis.''  Thus,  in  fluxions 
and  the  method  of  limits,  x  is  increased  by  /,  and, 
in  the  case  of  x'"^   the  increment   of  the   function, 

divided  by  /,  is  inx"'~^-\ — ^^ ^;ir'"~VH-,  etc.  ;  then, 

z 

putting,  /=o,  there  results  mx"'~^.  But  since  the 
expansion  of  {x-^-i)'"  was  effected  '*on  the  express 
supposition,  that  i  is  some  quantity,  if  you  take 
/=o,  the  hypothesis  is,  as  Berkeley  says,  shifted, 
and  there  is  a  manifest  sophism  in  the  process  " 
(p.  xii). 


266  LIMITS    AND   FLUXIONS 

227.  As  another  objection  to  limits,  or  prime 
and  ultimate  ratios,  Woodhouse  declares  that  *  '  the 
method  is  not  perspicuous,  inasmuch  as  it  considers 
quantities  in  the  state,  in  which  they  cease  to  be 
quantities." 

Moreover,  "the  definition  of  a  limit,  is  neither 
simple  nor  concise"  (p.  xvii).  '  '  The  name  of  Berkeley 
has  occurred  more  than  once  in  the  preceding  pages  : 
and  I  cannot  quit  this  part  of  my  subject  without 
commending  the  Analyst  and  the  subsequent  pieces, 
as  formlng  the  most  satisfactory  controversial  dis- 
cussion  in  pure  science,  that  ever  yet  appeared  :  into 
what  perfection  of  perspicuity  and  of  logicai  pre- 
cision,  the  doctrine  of  fluxions  may  be  advanced,  is 
no  subject  of  consideration  :  But,  view  the  doctrine 
as  Berkeley  found  it,  and  its  defects  in  metaphysics 
and  logie  are  clearjy  made  out.  If,  for  the  purpose 
of  habituating  the  mind  to  just  reasoning  ...  I 
were  to  recommend  a  book,  it  should  be  the  Analyst.'' 
"The  most  diffuse  and  celebrated  antagonists  of 
Berkeley,  are  Maclaurin  and  Robins,  men  of  great 
knowledge  and  sagacity  :  but  the  prolixity  of  their 
reasonings  confirms  the  notion,  that  the  method  they 
defend  is  an  incommodious  one.  " 

"  Landen,  I  believe,  first  considered  and  proposed 
to  treat  the  fluxionary  calculus  merely  as  a  branch  of 
Algebra  :  After  him,  M.  Lagrange,  a  name  ever  to 
be  celebrated,  in  the  Berlin  Acts  for  1772,  laid 
down  its  analytical  principles  ;  and  subsequently  in 
his  Théoric  des  fo7ictio)is  mialytiques^  1796,  he  has 
resumed   the    subject  :    in  this   treatise,   the   author 


CRITICISMS   BY  BRTTISH    WRITERS     267 

expressly  proposes,  to  lay  down  the  principles  of  the 
differential  calculus,  independently  of  ali  considera- 
tion  of  infinitely  small,  or  vanishing  quantities,  of 
limits,  or  of  fluxions  "  (p.  xviii).  While  Wood- 
house  considers  Lagrange's  discussion  as  very  valu- 
able,  he  does  not  find  it  free  from  logicai  faults. 

William  Hales,  1804 

228.  As  a  protest  against  the  new  movement  and 
a  vindication  of  Newton  from  the  attacks  upon 
fluxions  in  the  Monthly  Review^  William  Hales  pre- 
pared  a  hook,  the  Analysis  F/uxionum,  which  was 
published  in  Maseres'  Scriptores  Logarithniici^  voi.  v, 
London,  1804.  Hales  endeavours  to  show  that  the 
doctrine  of  prime  and  ultimate  ratios  is  really  the 
same  as  the  doctrine  of  the  limits  of  the  ratios. 
Hales's  fundamental  definitions  are  : 

''  Rationes  ultimai  sunt  limites,  ad  quos  quanti- 
tatum  sine  fine  decrescentium  rationes,  i,  semper 
appropinquant  ;  et,  2,  quas  propiùs  assequi  possunt 
quàm  prò  data  quàvis  differentià  ;  3,  nunquam  vero 
transgredi  ;  4,  nec  priùs  attingere,  quam  quantitates 
ipsai  diminuuntur  in  infinitum." 

"Momentum  est  fluentis  augmentum  aut  decre- 
meniuiìi  iìwinentaneuiii\  id  est,  tempore  quam  minimo 
genitum.      Estque  fluxioni  proportionale." 

After  Hales's  work  had  gone  to  press,  he  became 
acquainted  with  Benjamin  Robins's  Discourse  oi  1735, 
and  published  in  appendices^  numerous  extracts  from 

^  Maseres,  Scriptores  Logarif limici,  voi.  v,  pp.  848,  854,  856. 


268  LIMITS    AND   FLUXIONS 

it.  Says  Hales  :  "  It  is  far  superior  indeed  to  the 
subsequent  explanations  of  professed  commentators  ; 
and  it  is  a  high  gratification  to  myself  to  find,  that 
the  mode  of  explanation,  which  I  adopted  of  the 
Doctrine  of  Limits,  is  precisely  the  same  as  Robins's; 
long  before  1  had  seen  his  admirable  treatise,  which 
did  not  fall  into  my  hands  until  lately,  a  considerable 
time  after  the  publication  of  the  Afialysis  Fluxionum.'" 
Maseres  calls  the  Discourse  oi  Robins  ''the  ablest 
tract  that  has  ever  been  published  on  the  subject." 
Hales's  text  and  the  appendices  to  it  contain  con- 
siderable historical  material,  consisting  mainly  of 
references  to  and  quotations  from  earlier  writings. 
In  view  of  the  testimony  of  Laplace,  Legendre,  and 
Lacroix  on  the  superiority  of  the  method  of  fluxions, 
"  how  was  it  possible,"  asks  Hales,  that  the  eyes  of 
the  Monthly  Reviewers  **could  stili  be  so  holden 
.  .  .  as  stili  to  assert,  that  Newton  himself  was  not 
perfectly  satisfied  of  the  stability  of  the  ground  on 
which  he  had  established  his  Method  of  Fluxions  !  " 
Hales's  motive  in  opposing  Continental  ideas  was 
probably  partly  theological.  D'Alembert,  con- 
sidered  by  him  a  hostile  critic  of  Newton,  is  called 
*'  a  philosophizing  infidel,"  one  "  of  the  originai  con- 
spirators  against  Christianity,"  **at  once  the  glory 
and  disgrace  of  the  French  Academy  of  Sciences," 
whose  last  words  were  *'a  terrific  contrast  to 
the  death  of  the  Christian  Philosopher,"  Colin 
Maclaurin.^ 

The    publication    of    Hales's    Fluxions    in    large 

^  Maseres,  Scriptores  Logarithmici^  voi.  v,  pp.  176-182. 


CRITICI  SMS   BY  BRITISH    WRITERS     269 

quarto  form  and  in  the  Latin  language,  the  in- 
clusion  in  the  Appendix  of  matters  foreign  to  the 
subject  of  the  hook,  together  with  the  attempt 
to  maintain  a  system  of  notation  and  mode  of 
exposition  that  was  beginning  to  be  considered 
provincial,  caused  the  hook  to  ''fall  still-born 
from  the  press." 

Encyclopcedia  Britannica,   18 io 

229.  In  the  fourth  edition  of  the  Encydopcsdia 
Britannica,  Edinburgh,  18 io,  the  article  "Fluxions" 
is  wholly  rewritten,  and  is  much  more  extensive  than 
the  article  in  former  editions.  There  is  a  lengthy 
historical  introduction,  and  emphasis  is  placed  upon 
work  done  on  the  Continent.  It  observes  "that 
there  is  no  work  in  the  English  language  that  ex- 
hibits  a  complete  view  of  the  theory  of  fluxions, 
with  ali  the  improvements  that  bave  been  made 
upon  it  to  the  present  time."  Mention  is  made 
then  of  "several  excellent  works  in  the  French 
language,"  mentioning  Cousin,  Bossut,  La  Croix, 
L'Huilier. 

Letting  «  be  "  any  function  "  of  x,  the  //;;///  of 
the  ratio  {u  —u)  \  h  is  defined  as  "a  quantity  to 
which  the  ratio  may  approach  nearer  than  by  any 
assignable  difìference,  but  to  which  it  cannot  be  con- 
sidered as  becoming  absolutely  equal."  The  article 
asserts  that  the  method  of  fluxions  "  rests  upon  a 
principle  purely  analytical,  namely  the  theory  of 
limiting  ratios  ;  and  this  being  the  case,  the  subject 
may  be  treated  as  a  branch  of  pure  mathematics, 


270  LIMITS   AND   FLUXIONS 

without  having  occasion  to  introduce  any  ideas 
foreign  to  geometry.  Sir  Isaac  Newton,  however, 
in  first  delivering  the  principles  of  the  method, 
thought  proper  to  employ  considerations  drawn  from 
the  theory  of  motion.  But  he  appears  to  have  done 
this  chiefly  for  the  purpose  of  illustration,  for  he 
immediately  has  recourse  to  the  theory  of  limiting 
ratios,  and  it  has  been  the  opinion  of  several  mathe- 
maticians  of  great  eminence  (such  as  Lagrange, 
Cousin,  La  Croix,  etc. ,  abroad,  and  Landen  in  this 
country)  that  the  consideration  of  motion  was  intro- 
duced  into  the  method  of  fluxions  at  first  without 
necessity,  and  that  succeeding  writers  on  the  subject 
ought  to  have  estabhshed  the  theory  upon  principles 
purely  mathematica!,  independent  of  the  ideas  of 
time  and  velocity,  both  of  which  seem  foreign  to 
investigations  relating  to  abstract  quantity."  **By 
the  fluxions  then  of  two  variable  quantities  having 
any  assigned  relation  to  each  other,  we  are  in  the 
foUowing  treatise  always  to  be  understood  to  mean 
any  indefi^tite  quantities  which  have  to  each  other  the 
limiting  ratio  of  their  simultaneous  inci'cjuents  (we 
.  .  .  mean  the  ratio  of  the  nuinerical  values  of  the 
increments,  which  may  always  be  compared  with 
each  other,  whether  the  variable  quantities  be  of 
the  same  kind,  as  both  lines,  or  both  surfaces,  etc, 
or  of  different  kinds,  as  the  one  a  line,  and  the  other 
a  surface).  The  Newtonian  notation  is  used  in  the 
article  exclusively." 


CRITICI  SMS   BY  BRITISH    WRITERS     271 

Lacroix's   '■^  Elementary   Treatise,''    18 16 

230.  The  translation  of  Lacroix's  Elementary 
Trenti  se  on  the  Differential  and  Integrai  Calculus  ^ 
in  18 16  marks  an  important  period  of  transition. 

From  the  '*  Advertisement  "  we  quote  : 

This  work  of  Lacroix  ''  may  be  considered  as  an 
abridgement  of  his  great  work  on  the  Differential 
and  Integrai  Calculus,  although  in  the  demonstra- 
tion  of  the  first  principles,  he  has  substituted  the 
method  of  limits  of  D'Alembert,  in  the  place  of  the 
more  correct  and  naturai  method  of  Lagrange, 
which  was  adopted  in  the  former.  The  first  part 
of  this  Treatise,  which  is  devoted  to  the  exposition 
of  the  principles  of  the  Differential  Calculus,  was 
translated  by  Mr.  Babbage.  The  translation  of  the 
second  part,  which  treats  of  the  Integrai  Calculus, 
was  executed  by  Mr.  G.  Peacock,  of  Trinity  College, 
and  by  Mr.  Herschel,  of  St.  John's  College,  in  nearly 
equal  proportions." 

On  p.  2  the  process  of  differentiation  of  u^ax^ 
is  explained,  so  that  2ax  "  is  the  Hmit  "  of  the  ratio 
{u' —  u)  I  h,  or  it  is  **the  value  towards  which 
this  ratio  tends  in  proportion  as  the  quantity  // 
diminishes,  and  to  which  it  may  approach  as  near 
as  we  choose  to  make  it.  " 

Thus  Lacroix's  definition,  like  D'Alembert's,  does 
not  prohibit  the  limit  to  be  reached.  In  Note  A, 
added  by  the  translators,  we  read  : 

^  Alt  Elementary  Treatise  on  the  Diffei  ential  and  Integrai  Calculus. 
By  S.  F.  Lacroix.     Translated  from  me  French.     Cambridge.  1816. 


272  LIMirS    AND   FLUXIONS 

**  A  limit,  according  to  the  notions  of  the 
ancients,  is  some  fixed  quantity,  to  which  another 
of  variable  magnitude  can  never  become  equal, 
though  in  the  course  of  its  variation  it  may  approach 
nearer  to  it  than  any  difìference  that  can  be 
assigned.  "  Thus,  the  method  of  limits  is  bere 
ascribed  by  the  translators  to  the  ancients,  which 
is  an  act  of  reading  into  the  ancient  expositions  a 
theory  not  actually  there.  The  ancient  *  '  Method 
of  Exhaustions  "  is  merely  a  prelude  to  the  theory 
of  limits.  Peacock  gives  in  Note  A  a  history  of 
the  theory  of  limits,  in  which  researches  on  the 
Continent  are  dwelled  upon  and  the  contribution 
made  by  Newton  is  explained,  but  no  reference  is 
made  to  Jurin,  Robins,  and  Maclaurin.  In  Note  B 
Peacock  states  that  the  method  used  by  Lacroix  in 
this  treatise  "  was  first  given  by  D'Alembert,  in 
\}[i^  Encyclopédie'' z.xWq\q  '*  Différentiel.  "  Evidently 
Peacock  was  not  altogether  friendly  toward  this 
method,  for  in  Note  B  he  proceeds  "  directly  to 
show  in  what  manner  this  calculus  may  be  estab- 
lished  upon  principles  which  are  entirely  indepen- 
dent  of  infinitesimals  or  limits,"  and  then  informs 
the  reader  **that  we  are  indebted  for  the  principal 
part  of  the  contents  of  this  note,  to  the  Calcul  des 
P'onctions  of  Lagrange  and  the  large  treatise  by  our 
author,  on  the  Differential  and  Integrai  Calculus." 
Peacock  proceeds  to  give  an  account  of  Lagrange's 
calculus  of  functions  and  of  the  method  of  fluxions. 
Attention  is  called  to  ''the  difìfìculty  of  denoting 
the   operations   of   finding    the    difìferent    orders   of 


CRITICISMS  BY  BRITISH    WRITERS     273 

fluxions  "  according  to  the  Newtonian  notation, 
"when  for  u  we  put  the  function  itself,  which  it 
represents.  " 

23 1.  The  attitude  of  some  British  mathematicians 
of  the  early  part  of  the  nineteenth  century  toward 
the  discussions  of  the  fundamental  concepts  of  the 
calculus  carried  on  during  the  eighteenth  century 
is  exhibited  in  the  foUowmg  passage  from  John 
Leslie's  Dissertation  on  the  progress  of  mathematica! 
and  physical  science  :  ^ 

*'The  notion  of  flowing  quantities,  .  .  .  appears 
on  the  whole,  very  clear  and  satisfactory  ;  nor 
should  the  metaphysical  objection  of  introducing 
ideas  of  motion  into  Geometry  have  much  weight. 
Maclaurin  was  induced,  however,  by  such  cavelling, 
to  devote  half  a  volume  to  an  able  but  superfluous 
discussion  of  this  question.  As  a  refinement  on  the 
ancient  process  of  Exhaustions,  the  noted  method 
of  Prime  and  Ultimate  Ratios  .  .  .  deserves  the 
highest  praise  for  accuracy  of  conception.  It  has 
been  justly  commended  by  D'Alembert,  who  ex- 
pounded  it  copiously,  and  adapted  it  as  the  basis  of 
the  Higher  Calculus.  The  same  doctrine  was  like- 
wise  elucidated  by  our  acute  countryman  Robins  ; 
.  .  .  Landen,  one  of  those  men  so  frequent  in 
England  whose  talents  surmount  their  narrow 
education,  produced  in  1758,  a  new  form  of  the 
Fluxionary  Calculus,  under  the  title  of  Residuai 
Analysis,  which,  though  framed  with  little  elegance, 

^  Dissertation    Fourth,    in   the   Encyclopcedia   Britannica^   7th  ed., 
voi.  i,   1842,   pp.  600,  601. 

18 


274  LIMITS    AND   FLUXIONS 

may  be  deemed,  on  the  whole,  an  improvement  on  the 
method  of  ultimate  ratios.  " 

Remar ks 
232.  The  first  part  of  the  nineteenth  century 
marks  a  turning-point  in  the  study  and  teaching  of 
mathematics  in  Great  Britain.  i\ttention  has  been 
directed  to  the  efforts  of  Woodhouse  to  introduce 
the  higher  analysis  of  the  BernoulHs,  Euler,  Clairaut, 
and  Lagrange.  His  efforts  were  strongly  and  ably 
seconded  by  three  other  young  men  at  Cambridge, 
John  Frederick  William  Herschel,  Charles  Babbage, 
and  George  Peacock,  who  used  to  breakfast  together 
on  Sunday  mornings,  and  in  18 12  founded  the 
"  Analytical  Society  at  Cambridge,"  for  the  promo- 
tion, as  Babbage  humorously  expressed  it,  of  *'  the 
principles  of  pure  D-ism  in  opposition  to  the  Dot-digo. 
of  the  University."  The  translation  into  English  of 
Lacroix's  Elementary  Treatise  and  the  publication, 
in  1820,  of  Exaniples  with  their  solutions,  brought 
the  more  perfect  notation  of  Leibniz  and  the  re- 
fined  analytical  methods  to  the  attention  of  young 
students  of  mathematics  in  England.^ 

^  Before  the  nineteenth  century,  the  use  in  England  of  the  Leibnizian 
notation  dz  and  Jydz  is  exceedingly  rare.  In  our  §  54  we  saw  that 
about  the  beginning  of  the  eighteenth  century  these  symbols  were  used 
by  John  Craig  in  articles  published  in  the  London  Philosophical 
Transactions.  When  criticising  Euler,  Benjamin  Robins  once  used 
the  Leibnizian  notation  ;  see  our  §  142.  Mr.  Philip  E.  B,  Jourdain  has 
brought  to  my  attention  the  fact  that  the  sign  of  integration  f  occurs 
also  in  a  hook,  entitied,  Second  Volume  of  ihe  Instrtutions  given  in 
the  Drawitt^  School  established  by  the  Dtiblin  Society.  .  .  .  Under  the 
Direction  of  [oseph  Fenti,  heretofore  Professor  of  Philosophy  in  the 
University  of  Nantes.  Bub/in,  UDCCLXXll.  De  Morgan  refers  to 
this  work  in  a  letter  to  Hamilton.  See  Graves'  Life  of  Sir  William 
Rowan  Hamilton^  voi.  iii,  p.  488.     See  also  our  Addenda^  p.  2S9. 


CRTTICISMS   BY  BRITISH    WRITERS     275 

As  usually  happens  in  reformations,  so  here,  some 
meritorious  features  were  discarded  along  with  what 
was  antiquated.  William  Hales,  in  1804,  referred 
to  the  much  neglected  Discourse  of  Benjamin  Robins 
(i735)>  with  its  full  and  complete  disavowal  of 
infinitesimals  and  clear-cut,  though  narrow,  con- 
ception  of  a  limit.  By  a  curious  turn  in  the 
process  of  events,  Robins  was  quite  forgotten  in 
England,  and  D'Alembert's  definition  was  recom- 
mended  and  widely  used  in  England.  Now  Robins 
and  D'Alembert  had  the  same  conception  of  a  limit  ; 
both  held  to  the  view  that  variables  cannot  reach 
their  limits.  However,  there  was  one  difìference 
between  the  two  men:  Robins  embodied  this  restric- 
tion  in  his  definition  of  a  limit  ;  D'Alembert  omitted 
it  from  his  definition,  but  referred  to  it  in  his 
explanatory  remarks.      D'Alembert  says  :  ^ 

**  On  dit  qu'une  grandeur  est  la  limite  d'une  autre 
grandeur,  quand  la  seconde  peut  approcher  de  la 
première  plus  près  que  d'une  grandeur  donnée,  si 
petite  qu'on  là  puisse  supposer,  sans  pourtant  que  la 
grandeur  qui  approche,  puisse  jamais  surpasser  la 
grandeur  dont  elle  approche  ;  ensorte  que  la  différ- 
ence  d'une  pareille  quantité  à  la  limite  est  absolu- 
ment  inassignable."  Further  on  in  the  same  article 
we  read  :  "A  proprement  parler,  la  limite  ne  co- 
incide jamais,  ou  ne  devient  jamais  égale  à  la 
quantité  dont  elle  est   la  limite  ;   mais  celle-ci   s'en 


^  Art.  "Limite"  in  the  Encyclopédie,  ou  dictionnaìre  rais  sonni'  des 
Sciences  des  arts  et  des  tnétiers,  puh  li é  par  M.  Diderot,  et  M.  D  Alembert, 
Paris,  1754.     See  also  the  later  edition  of  Geneva,  1772. 


276  LIMITS   AND   FLUXIONS 

approche  toujours  de  plus  en  plus,  &  peut  en  différer 
aussi  peu  qu'on  voudra.  .  .  .  On  dit  que  la  somme 
d'une  progression  géométrique  décroissante  dont  le 
premier  terme  est  <^  &  le  second  b,  est  (a  —  ò)  /  (aa)  ; 
cette  valeur  n'est  poit  proprement  la  somme  de  la 
progression,  c'est  la  limite  de  cette  somme,  c'est-à- 
dire  la  quantité  dont  elle  peut  approcher  si  près 
qu'on  voudra,  sans  jamais  y  arriver  exactement." 

233.  That  even  the  best  expositions  of  limits  and 
the  calculus  that  the  Continent  had  to  offer  at  that 
time  were  recognised  in  England  to  be  imperfect,  is 
shown  by  a  passage  in  a  letter  which  William  Rowan 
Hamilton  wrote  in  1828  to  his  friend  John  T. 
Graves  :  ^ 

**I  have  always  been  greatly  dissatisfied  with 
the  phrases,  if  not  the  reasonings,  of  even  very 
eminent  analysts,  on  a  variety  of  subjects.  .  .  .  An 
algebraist  who  should  thus  clear  away  the  meta- 
physical  stumbling-blocks  that  beset  the  entrance 
of  analysis,  without  sacrificing  those  concise  and 
powerful  methods  which  constitute  its  essence  and 
its  value,  would  perform  a  useful  work  and  deserve 
well  of  Science." 

^  Life  of  Siy  William  Rowan  Hamilton,  by  Robert  P.  Graves,  voi.  i, 
1882,  p.  304. 


CHAPTER    XII 

MERITS    AND     DEFECTS    OF    THE    EIGHTEENTH- 
CENTURY    BRITISH    FLUXIONAL   CONCEPTIONS 

Merits 
234.  There  are,  perhaps,  no  intuitional  concep- 
tions  available  in  the  study  of  the  calculus  which 
are  clearer  and  sharper  than  motion  and  velocity. 
There  is,  therefore,  a  certain  advantage  in  approach- 
ing  the  first  study  of  the  difìferential  calculus  or  of 
fluxions  by  the  consideration  of  motion  and  velocity. 
Even  in  modem  teaching  of  the  elements  to 
beginners,  we  cannot  afìford  to  ignore  this  advantage 
offered  by  the  eighteenth-century  British  mode  of 
treating  the  calculus. 

A  second  point  of  merit  lies  in  the  abandonment 
of  the  use  of  infinitely  little  quantities.  Not  ali 
English  authors  of  the  eighteenth  century  broke 
away  from  infinitesimals,  but  those  who  did  vvere 
among  the  leaders  :  Robins,  Maclaurin,  Simpson, 
Vince,  and  a  few  others.  The  existence  of  infini- 
tesimals (defined  as  infinitely  small  constants)  was 
looked  upon  by  philosophers  and  by  many  mathe- 
maticians  as  doubtful.  Their  subjective  existence 
was  hardly  more  probable  than  their  objective  exist- 
ence.    These    mystic   creations   occupied    a   hypo- 

277 


278  LIMITS    AND   FLUXIONS 

thetical  twilight  zone  between  finite  quantity  and 
no  quantity.  Their  abandonment  added  to  the 
clearness  and  logicai  rigour  of  mathematics.  From 
the  standpoint  of  rigour,  the  British  treatment  of 
the  calculus  was  far  in  advance  of  the  Continental. 
It  is  certainly  remarkable  that  in  Great  Britain 
there  was  achieved  in  the  eighteenth  century,  in  the 
geometrical  treatment  of  fluxions,  that  which  was 
not  achieved  in  the  algebraical  treatment  until  the 
nineteenth  century  ;  it  was  not  until  after  the  time 
of  Weierstrass  that  infinitesimals  were  cast  aside 
by  many  mathematical  writers  on  the  Continent. 

235.  There  is  a  perversity  in  historic  events 
exhibited  in  the  fact  that  after  infinitesimals  had 
been  largely  expelled  in  the  eighteenth  century 
from  Great  Britain  as  undesirable,  unreal,  and 
mischief-making,  they  should  in  the  nineteenth 
century  be  permitted  to  return  again  and  to  flourish 
for  a  time  as  never  before.  About  18 16  the 
Leibnizian  notation  of  the  calculus  and  the  vast 
treasures  of  mathematical  analysis  due  to  the 
Bernoullis,  Euler,  D'Alembert,  Clairaut,  Lagrange, 
Laplace,  Legendre,  and  others,  which  were  ali  ex- 
pressed  in  that  notation,  found  their  way  into 
England.  This  influx  led  to  enrichment  and  advance- 
ment  of  mathematics  in  England,  but  also  to  a 
recrudescence — this  return  of  the  infinitely  small. 
How  thoroughly  the  infinitesimal  invaded  certain 
parts  of  British  territory  is  seen  in  Frice's  large 
work  on  the  Infinitesiinal  Calculus^  a  work  which  in 
many  ways  is  most  admirablc  and  useful. 


MERITS   AND   DEFECTS  279 

236.  After  the  development  of  the  theory  of 
limits  by  the  English  mathematicians  and  by  such 
Continental  writers  as  D'Alembert  and  Lacroix,  it 
would  hardly  seem  necessary  even  for  the  sake  of 
brevity  to  reintroduce  the  old-time  infinitesimal 
which  could  be  *'dropped"  whenever  it  was  very 
small,  yet  stood  in  the  way.  But  at  ali  times,  and 
particularly  in  the  eighteenth  and  beginning  of  the 
nineteenth  centuries,  there  have  been  mathematicians 
who  cared  little  for  the  logicai  foundations  of  their 
science.  Fascinated  by  the  ease  with  which  the 
calculus  enabled  them  to  dispose  of  difficult  prob- 
lems  in  the  theory  of  curves,  ordinary  mechanics, 
and  celestial  motions,  they  felt  more  like  poets,  and 
held  the  sentiments  toward  logie  that  a  distinguished 
hard  entertained  toward  pure  intellectualism  when 
he  contemplated  the  beauties  of  the  rainbow  : 

"  Triumphal  arch  that  fiU'st  the  sky, 
When  storms  prepare  to  part, 
I  ask  not  proud  philosophy 
To  teach  me  what  thou  art." 

Defects 

237.  AH  the  eighteenth-century  expositions  of 
the  foundations  of  the  calculus — even  the  British — 
are  defective.  Without  attempting  an  historical 
treatment  or  a  logicai  exposition  of  later  develop- 
ments,  we  desire  to  point  out  briefly  what  some  of 
these  defects  were. 

In  the  first  place,  the  doctrine  of  fluxions  was  so 
closely  associated  with  geometry,  to  the  neglect  of 


28o  LIMITS    AND   FLUXIONS 

analysis,  that,  apparently,  certain  British  writers 
held  the  view  that  fluxions  were  a  branch  of  geo- 
metry.  In  the  preface  to  the  GentlemarCs  Diary  of 
London,  the  new  editor,  Mr  Wildbore,  said  at  the 
commencement  of  his  editorship  in  1781,  *'the 
doctrine  of  fluxions  depends  on  principles  purely 
geometrical,  as  is  very  satisfactorily  demonstrated 
by  that  incomparable  geometer,  the  late  Dr  Robert 
Simson  of  Glasgow  in  his  Opera  posthuma.  " 

In  the  second  place,  as  pointed  out  by  Landen 
and  Woodhouse,  there  was  an  unnaturalness  in 
founding  the  calculus  upon  the  notions  of  motion 
and  velocity.  In  a  real  way,  these  notions  seem 
to  apply  only  to  a  limited  field  in  the  applications  of 
the  calculus,  namely,  to  dynamics.  In  other  fields, 
motion  and  velocity  are  wholly  foreign  concepts 
which,  if  applicable  at  ali,  are  so  only  in  a  figurative 
sense. 

238.  Newtonian  writers  lay  great  stress  upon 
such  conceptions  as  a  line  generated  by  the  motion 
of  a  point,  a  surface  generated  by  the  motion  of  a 
line,  and  a  solid  generated  by  the  motion  of  a 
surface.  We  have  already  referred  to  the  pedago- 
gica! advantages  of  this  view,  in  teaching  beginners. 
But  as  a  final  logicai  foundation  this  view  is  inade- 
quate. Not  ali  continuous  curves  can  be  conceived 
as  traceable  by  the  motion  of  a  point.  An  example 
frequently  quoted,  in  discussions  of  this  sort,  is 
the  curve 

y  —  0  for  ;r  =  o,  y  =  x  sin      for  x  =^  0. 


MERITS   AND  DEFECTS  281 

Let  US  try  to  trace  this  curve  by  the  motion  of  a 
point  starting  from  the  origin  of  co-ordinates.  In 
which  direction  must  the  point  move  from  the 
origin  ?  To  answer  this  question  we  differentiate, 
and  find  dy  j  dx=s\n  {i  I  x)  —  {i  /  x)  cos  ( i  /  x).  At 
the  origin  we  have  x  =  o  and  7  =  0.  No  value  can  be 
assigned  to  dy  /  dx,  because  i  /  x  has  no  meaning 
when^  =  o  ;  moreover,the  equationjr  =  ^' sin  (i  /  x)  ìs 
expressly  stated  above  to  apply  only  when  x  is  not 
zero.  There  is,  therefore,  no  way  of  ascertaining 
the  direction  in  which  the  point  must  depart  from 
the  origin.  Perhaps  we  can  do  better  if  the  moving 
point  is  started  at  another  part  of  the  curve.  An 
attempt  to  plot  the  curve  reveals  the  fact  that  it 
lies  between  two  right  Hnes,  of  which  one  makes 
with  the  ;i'-axis  an  angle  of  45°,  the  other  an  angle 
of  —45°.  As  the  point  moves  along  the  curve 
toward  the  origin,  the  curve  is  found  to  oscillate 
with  ever-increasing  rapidity.  When  we  try  to 
determine  the  direction  by  which  it  jumps  into  the 
origin,  we  encounter  the  same  difficulty  as  before. 
As  long  as  x  is  finite,  the  direction  of  motion  is 
determinable.  But  as  soon  as  we  try  x=o,  the 
determination  is  impossible.  This  conclusion  must 
be  accepted,  in  spite  of  the  fact  that  the  curve  is 
contiiiuous  in  ali  its  parts,  including  the  origin. 
This  example  illustrates  the  inadequacy  of  motion 
as  a  fundamental  concept. 

239.  Difficulties  are  encountered  in  the  notion  of 
velocity.  Is  variable  velocity  an  objective  reality  ? 
Take  a  body  falling  from   rest.      We   say  that  its 


282  LIMITS    AND   FLUXIONS 

velocity   is    ds  /  dt=^gt.      At   the    end   of  the    first 
second,  the  velocity  is  ^^.      If  we  ask  ourselves,  How 
far  does  the   body  move  with   the  velocity  gì  we 
must  admit  that  no  distance  can  be  assigned.      We 
cannot  say  that  the  body  moves  from  a  certain  point 
to  the  point  immediately  beneath  ;  there  is  no  such 
point  immediately  beneath.      For,  as  soon  as  we  try 
to  locate  such  a  point,  it  occurs  to  us  that  we  can 
imagine  at  least  one  point  located  between  the  two 
points  under  consideration.     This  intermediate  point 
serves   our   purposes  no   better,   for  a  fourth   point 
located  between   it   and  the  initial   point   is  easily 
detected,  and  so  on,  without  end.     Thus  it  is  seen 
that   no  distance,   however  small,   can  be  assigned 
through  which  a  body  falls  with  a  given  velocity. 
We  are  thus  compelled  to  reject  variable  velocity 
as  a   physical    fact.      What,    then,    is    ds  j  dt=gtì 
Clearly  it  is  merely  a  limit,  a  mathematical  concept, 
useful  in  mathematical  analysis,  but  without  physical 
reality.      To  say  that  ds  /  dt  represents  the  distance 
a  body   would  fall   in   unit   time  after  the  instant 
indicated  by  /,   is  to  assign  it  merely  hypothetical 
meaning,    destitute  of  concreteness.      While    these 
considerations  in  themselves  may  not  debar  the  use 
of  velocity  as  a  mathematical  concept  upon  which  to 
build  the  calculus,  they  show  that  the  concept  is  not 
as  simple  as  it  would  seem  to  be  at  first  approach. 

The  reader  will  have  observed  that  in  ali  discus- 
sion  of  limits  during  the  eighteenth  century  the 
question  of  the  existence  of  a  limit  of  a  convergent 
sequence  was  never  raised  ;  no  proof  was  ever  given 


MERITS  AND  DEFECTS  283 

that  a   limit   actually  exists.      In   this   respect   the 
treatment  vvas  purely  intuitive. 

240.  Another  defect  in  the  British  exposition  of 

fluxions  was   in  the   use   of  the  word    ''quantity.  " 

No  definition  of  it  was  given,   yet  quantities  were 

added,   subtracted,   multiplied,   and  divided.      It   is 

possible  to  treat  quantities  or  magnitudes  without  the 

use  of  number.     The  fifth  and  tenth  books  of  Euclid's 

Elenients  contain  such  treatment.     We  may  speak  of 

the  ratio  of  one  magnitude  to  another  magnitude,  or 

we  may  speak  of  the  ratio  of  one  number  to  another 

number.      Which    was    meant    in    the    treatment   of 

fluxions  ?     Straight  Hnes  were  drawn  and  the  ratios 

of  parts  of  these  lines  were  written  down.      What 

were   these   the   ratios   of?     Were   they  the   ratios 

of  the   Hne-segments  themselves,    or  the    ratios  of 

the  numbers  measuring   the   lengths  of  these  line- 

segments  ?     No  explicit  anewer  to  this  was  given. 

Our  understanding  of  authors  like  Maclaurin,  Rowe, 

and  others  is  that  in  initial  discussions  such  phrases 

as    *' fluxions  of  curvih'neal  figures,"   ''fluxion  of  a 

rectangle,"  are    used   in   a   non-arithmetical    sense  ; 

the   idea   is   purely   geometrical.      When    later    the 

finding  of  the  fluxions  of  terms  in  the  equations  of 

curves  is  taken  up,   the  arithmetical  or  algebraical 

conception  is  predominant.      Rarely  does  a  writer 

speak  of  the  difference  between  the  two.      Perhaps 

"  His  notions  fitted  things  so  well, 
That  which  was  which  he  could  not  teli." 

241.  Analytical    geometry    practically    identìfied 
geometry  with  arithmetic.      It  was  tacitly  assumed 


284  LIMITS   AND  FLUXIONS 

that  to  every  distance  corresponds  a  number  and  to 
every  number  there  corresponds  a  distance.  Number 
was  thus  given  a  geometrical  basis.  This  situation 
continued  into  the  nineteenth  century.  This  metrical 
view  involved  the  entire  theory  of  measurement, 
which  assumed  greater  difficulties  with  the  advent 
of  the  non-EucHdean  geometries.  The  geometrical 
theory  of  number  became  less  and  less  satisfactory 
as  a  logicai  foundation.  Hence  the  attempts  to 
construct  purely  arithmetical  theories.^ 

A  good  share  of  those  difficulties  arose  from 
irrational  numbers,  which  could  not  be  avoided  in 
analytical  geometry.  This  occurrence  is  not  merely 
occasionai  ;  irrational  ratios  are  at  least  as  frequent 
as  rational  ones.  What  is  an  irrational  number  ? 
How  do  we  operate  with  irrational  numbers  ?  What 
constitutes  the  sum,  difìference,  product  or  quotient, 
when  irrational  numbers«are  involved?  No  explicit 
answer  was  given  to  these  questions.  It  was  tacitly 
assumed  without  fear,  that  it  is  safe  to  operate  with 
irrational  numbers  as  if  they  were  rational.  But 
such  assumptions  are  dangerous.  They  might  lead 
to  absurdities.  Even  if  they  do  not,  this  matter 
demands  attention  when  mathematical  rigour  is 
the  aim. 

242.  Perhaps  it  may  be  worth  while  to  recali  to 
the  reader's  mind  illustrations  of  the  danger  result- 
ing  from  taking  operations  known  to  yield  consistent 

^  For  a  historical  account  of  the  number  concept  and  the  founding 
of  the  theory  of  transfinite  numbers  during  the  nineteenth  century, 
read  Philip  E.  B.  Jourdain's  "  Introduction  "  to  Cantor's  Transjinite 
Numbers,  The  Open  Court  Publishing  Co,,  191 5. 


MERITS  AND  DEFECTS  285 

results  when  a  certain  limited  class  of  numbers  is 
involved,  and  applying  them  to  numbers  of  a  more 
general  class.  Suppose  a  and  b  to  be  rational, 
positive  numbers,  not  zero  ;  we  find,  let  us  agree, 
consistent  results  in  the  operation  a-\-by  a  —  b  when 
a>by  and  av^b,  and  a-^b.  Let  us  now  consider 
the  class  composed  of  rational  numbers,  both 
positive  and  negative  ;  suppose,  moreover,  that 
we  introduce  o  in  order  to  give  interpretation  to 
the  operation  a  — a.  If  in  this  extended  class  of 
numbers  we  admit  the  four  operations  a-\-by  a  —  by 
axb,  a-^by  trouble  arises  even  after  due  considera- 
tion  has  been  given  to  the  negative  numbers.  There 
may  arise  the  following  well-known  paradox.  Let 
a  —  b=\y  then  a^  —  b'^  =  a  —  b.  Divide  both  sides  of 
the  last  equation  by  a  —  b^  and  we  have  a-\-b=i,  or 
2=1.  Where  is  the  diffìculty  ?  The  answer  is 
known  to  every  schoolboy  :  We  have  used  a  —  b^  or 
o,  as  a  divisor  ;  we  have  extended  the  operation  of 
division  to  the  larger  class  of  numbers,  and  to 
zero,  without  first  assuring  ourselves  that  such  an 
extension  is  possible  in  every  case  ;  division  by  zero 
is  inadmissible. 

243.  A  less  familiar  example  is  the  following. 
Let  US  suppose  that,  for  real  exponents,  it  is  estab- 
lished  that  ( A^)^  =  A'-^.  When  we  apply  this  process 
to  imaginary  exponents,  trouble  arises.  Take  the 
equation    e""*'""  —  e'^**'"\    where    in    and    n   are   distinct 

integers,  z'=  7- I,  7r=3'i4i59  •  •  .,  and  ^=2 7 18 
.    .    .      That  this  equation  holds  is  evident,  for  ^''"" 

=  cos  2?«7rH-/sin  2;;/7r  =  cos  2;/7r  +  /sin  2;/x  =  ^^''".      If 


286  LIMITS    AND   FLUXIONS 

both  sides  of  ^^"""  =  £* '"'  are  raised  to  the  power  i  /  2, 
vve  obtain  ^-""^  =  ^-«\  Here  ali  the  letters  stand  for 
real  numbers  ;  since  ;;/  and  n  are  not  equal  to  each 
other,  this  last  equation  is  an  absurdity.  The 
assumption  that  a  rule  of  operation  vab'd  for  real 
exponents  was  valid  also  for  imaginary  exponents, 
has  led  to  papable  error. 

Examples  of  this  sort  emphasise  the  need  of 
caution  when  operations,  known  to  be  valid  for  a 
certain  class  of  numbers,  are  applied  to  numbers 
belonging  to  a  larger  class.  Special  examination 
is  necessary.  These  remarks  are  pertinent  when 
operations  applicable  to  rational  numbers  are  ex- 
tended  to  a  class  which  embraces  both  rational  and 
irrational  numbers.  What  are  the  numbers  called 
irrational?  It  is  hardly  sufificient  to  say  that  an 
irrational  number  is  one  which  cannot  be  expressed 
as  the  ratio  of  two  rational  numbers.  A  negative 
definition  of  this  sort  does  not  even  establish 
the  existence  of  irrational  numbers.  Considerale 
attention  has  been  paid  to  the  definition  of  irrational 
numbers  as  limits  of  certain  sequences  of  rational 
numbers.  Thus,  ^2  may  be  looked  upon  as  the 
limit  of  the  sequence  of  rational  fractions  obtained 
by  the  ordinary  process  of  root-extraction,  namely, 
the  sequence,  i,  1*4,  1*41,  I'4I4,  I"4I42,  .  .  . 
This  attempt  to  establish  a  logicai  foundation 
for  irrational  numbers  was  not  successful.  We 
endeavour,  in  what  foUows,  to  make  this  matter 
plainer. 

244.    Let  US  agree  that   in  building  up  an  aiith- 


MERITS  AND  DEFECTS  287 

mctical  theory  vve  have  reached  a  development  of 
rational  numbers  (integers  and  rational  fractions). 
We  wish,  next,  to  define  //;////  and  also  irrntional 
number.  An  early  nineteenth-century  definition  of 
limit  was  :  '' VVhen  the  successive  values  attributed 
to  a  variable  approach  a  fixed  value  indefinitely  so 
as  to  end  by  difìfering  from  it  as  little  as  is  wished, 
this  fixed  value  is  called  the  limit  of  ali  the  others.  " 
Since,  according  to  our  supposition,  we  are  stili  in 
the  field  of  rational  numbers,  this  limit,  unless  it 
happens  to  involve  only  rational  numbers  and  to  be 
itself  only  a  rational  number,  is,  in  our  case,  non- 
existent  and  fictitious. 

If  now,  as  stated  above,  an  irrational  number  is 
defined  as  the  limit  of  certain  sequences  of  rational 
fractions,  trouble  arises.  The  existence  of  such  a 
Hmit  is  often  far  from  evident.  But  aside  from  that 
general  consideration,  the  difficulty  of  the  situation 
in  our  case  is  apparent  :  Irrational  numbers  are 
limits,  but  limits  themselves  are  non-existent  or 
fictitious,  unless  they  are  rational  numbers.  To 
avoid  this  breakdown  in  the  logicai  development, 
it  was  found  desirable  to  define  irrational  number 
without  using  limits. 

245.  Wìth  the  view  of  avoiding  the  use  of  limits 
in  the  definition  of  irrational  number,  and  at  the 
same  time  avoid  inelegant  and  difficult  assumptions, 
involving  complicated  considerations  relating  to  the 
nature  of  space, ^  devices  were  invented  by  several 

^  On  this  point  consult  the  article  "  Geometry  "  in  the  Encyclopitdia 
Britannica^  iith  edition,  the  part  on  Congruence  and  Measurement. 


288  LIMITS   AND   FLUXIONS 

logicians  independently,  which  freed  the  number  con- 
cept  from  magnitude  and  established  number  theory 
on  the  concept  of  order.  Chief  among  the  workers 
in  this  field  were  Méray,  Weierstrass,  Dedekind, 
and  Georg  Cantor.  It  is  to  them  that  we  owe  re- 
presentations  of  number,  both  rational  and  irrational, 
which  have  yielded  a  much  more  satisfactory  theory 
of  limits,  and  in  that  way  have  vastly  improved  the 
logicai  exposition  of  the  differential  calculus.  These 
theories  have  brought  about  the  last  stages  of  what 
is  called  the  aritJunetisation  of  mathematics.  As 
now  developed  in  books  which  aim  at  extreme 
rigour,  the  notion  of  a  limit  makes  no  reference  to 
quantity  and  is  a  purely  ordinai  notion.  Of  this 
mode  of  treatment  the  eighteenth  century  had  never 
dreamed. 


ADDENDA   TO    §§   54,    58,    73 

246.  Additional  data  on  the  fundamental  concep- 
tions  relating  to  fluxions  and  on  the  use  of  the 
Newtonian  and  the  Leibnizian  notations  in  England 
during  the  Hfetime  of  Newton  are  contained  in 
George  Cheyne's  Philosophical  Principles  of  Relzgioti, 
Part  li,  London,  17 16.  Fart  I  of  this  book 
appeared  first  in  1705.  Like  Berkeley's  Analyst, 
which  was  written  later,  Cheyne's  book,  Fart  I  and 
Fart  II,  had  for  its  primary  purpose  the  refutation 
of  atheism.  Cheyne  says  in  his  preface  to  the  third 
edition  of  Fart  I,  "that  Atheism,  may  be  eternally 
confounded,  by  the  most  distant  Approaches  to  the 
true  Causes  of  naturai  Appearances.  And  that  if 
the  Modem  Fhilosophy  demonstrates  nothing  else, 
yet  it  infallibly  proves  Atheism  to  be  the  most 
gross   Ignorance.  " 

Fart  1 1  of  Cheyne's  book  consists  of  three  chapters 
and  of  seven  pages  of  "  Additions.  "  He  says  in  his 
preface  to  this  part  that,  excepting  one  short  note, 
the  third  chapter  and  the  "Additions"  are  "  what 
the  reverend  and  ingenious  Mr.  John  Craig  sent  me 
about  seven  years  ago,  when  1  desired  him  .  .  .  to 
vvrite  me  down  his  Thoughts  on,  correct  or  alter, 

2^9  19 


290  LIMITS  AND  FLUXIONS 

vvhat  I  had  formerly  published  on  this  Head  in  the 
first  Edition  of  this  Work." 

Cheyne  uses  in  Part  II,  p.  20,  the  notation  ±  io 
denote  a  distance  B^  when  he  supposes  "  b  infinitely 
near  to  B."  In  §  58  we  pointed  out  that  in  1704 
Cheyne  wrote  once  x=  i,  but  nowhere  in  the  present 
hook  does  x  denote  a  finite  quantity.  He  argues 
that  I  /  i'=  co  I,  that  i  /  o=co  ;  hence  thati-=^,  or 
"relative  nothing,"  which  is  **  the  least  Part  of  the 
Finite,  to  which  it  is  related  or  compared.  "  On 
p.  21  he  calls  x  "  an  infinitely  little  Part  of  x."  On 
p.  12  he  speaks  of  the  "  absolute  infinite"  as  "  ad- 
mitting  of  neither  Increase,  nor  Diminution,  or  of 
any  Operation  that  mathematica!  Quantity  is  sub- 
jected  to,"  while  (p.  13)  "  absolute  nothing"  is 
''neither  capable  of  increasing  nor  diminishing,  nor 
of  any  wise  altering  any  Mathematica!  Quantity  to 
which  it  is  apply'd,  but  stands  in  full  opposition  to 
absolute  Infinite. "  On  the  other band,  "indefinite" 
or  '  '  relative  infinite  "  quantities  (p.  29")  '  '  are  not 
properly  either  Finite  or  Infinite,  but  between  both.  " 
The  "  relative  nothing  "  (p.  8)  "  is  an  infinitely  little 
Quantity,  as  it  stands  related  to  a  given  Finite, 
by  the  perpetuai  Subtraction  of  which  from  it  self 
it  is  generated.  Let  0  stand  for  relative  nothing. 
Thus  <?i  is  a  relative  infinitely  little  Quantity,  as  it 
stands  related  to  Unity,  by  the  perpetuai  Subtrac- 
tion of  which  from  it  self,  it  is  generated  ;  that  is 
£?i  =  i  — i  +  i  — i  +  i  — i-f-i  — I  ec,  and  oa  is  an 
infinitely  little  Quantity,  as  it  stands  related  to 
the  given   F'niite  a^   by   the    perpetuai   Subtraction 


ADDENDA  291 

of  vvhich  from  it  self,  it  is  generateci  ;  that  is 
oa  —  a^a-\-a  —  a-Ya  —  a  ec."  In  the  "  Advertize- 
ment  "  foUowing  p.  190  this  is  further  explained 
thus  :  '  '  Relative  Nothing  is  said  here  to  be  generated 
by  a  perpetuai  Subtraction,  tho'  the  Signs  by  alter- 
nately  -f-  and  — .  For  these  Reasons,  because 
relative  Infinite,  was  said  to  be  generated  by  a 
perpetuai  Addition,  and  because  that  after  the  first 
Term,  every  two  succeeding  ones  in  relative  Nothing 
I  is  equivalent  to  o\  thus  i  —  i  +  i  —  i  +  i  —  i,  &c. 
.    .    .    =\—oi—o\—o\  &c." 

247.  These  explanations  are  intended  by  Cheyne 
merely  as  introductions  to  the  later  chapters,  par- 
ticularly  that  by  John  Craig,  who  (p.  167)  declares 
that  0  cannot  be  an  absolute  nothing,  "for  an 
infinite  Number  of  absolute  Nothings  cannot  make  i, 
but  by  0  is  understood  an  infinitely  small  part,  as 
in  the  cale.  diff.  dx  is  an  infinitely  small  part  of  x, 
so  that  dx  is  as  0  \.o  x  :  Not  that  dx  is  absolutely 
nothing,  for  it  is  divisible  into  an  infinite  Number  of 
Parts,  each  of  which  is  ddx.  "  To  make  the  point 
stili  plainer,  John  Craig  continues  (p.  168):  "But 
then  it  may  be  inquir'd  what  is  the  Ouotient  that 
ariscs  from  the  Division  of  i  by  absolute  Nothing. 
I  say  there  is  no  Ouotient  because  there  is  no 
Division  :  Therefore  it  is  a  Mistake  to  say  the 
Quotient  is  i  or  Unity  undivided,  which  is  demon- 
strably  false,  neither  is  the  Ouotient  =  ^.  For 
properly  speaking  there  is  no  Ouotient,  and  there- 
fore it  is  an  Error  to  assign  any.  In  like  manner, 
it  is  an  Error  to  say,  that  oy^a  makes  the  Product 


292  LIMITS  AND  FLUXIONS 

o  ;  for  properly  speaking  there  is  no  Product.  It  is 
true,  this  of  Multiplication  has  no  influence  upon 
Practice,  but  that  of  Division  has.  From  hence  it 
appears,  that  a  Curve  is  saicl  to  meet  with  its  Asymp- 
tote,  when  the  Ordinate  is  infinitely  little."  Then 
follows  a  startling  view  which  had  been  held  about 
sixty  years  before  by  John  Wallis  in  his  Arithmetica 
Infinitoruin,  1655,^  but  Craig  makes  no  reference  to 
him.  Craig  argues  (p.  169):  "This  same  Notion 
does  explain  how  it  comes  to  pass  that  i  divided  by 
a  negative  Number  gives  a  Ouotient  greater  than 
Infinite."  Curiously,  he  represents  the  logarithmic 
curve  j/  =  log;r  as  crossing  the  r-axis  at  y=  —00, 
for  since  the  curve  approaches  the  y  -  axis  in- 
finitely near  when  positive  x  approaches  zero,  '*we 
may  conceive  the  Logarithmic  Curve  continued  as 
intersecting  "  the  y-axis,  so  as  to  form  "  one  con- 
tinued Curve."  Accordingly  negative  numbers  have 
logarithms  that  are  real  and  negative.  His  further 
argument  amounts  to  this  :  For  values  of  x  that  are 
equal  to  i  divided  by  a  negative  number,  yìny  —  log jr 
is  negative  and  is  less  than  its  value  —  oo  arising 
when  x=  0  (presumably  in  the  sense  that  —  2  <  —  i). 
"  Ergo  ;ir  is  a  Number  greater  than  infinite."  Con- 
sidering  the  approach  of  the  logarithmic  curve 
towards  its  asymptote,  Craig  says  (p.  170)  that 
"bere  it  is  observable,  that  there  are  affirmative 
Numbers  less  than  nothing  denoted  by  the  several 
Powers  oidx,  as  dx'^,  dx^,  ec. ,  or  by  the  second,  third, 
ec.  Differences,  and  these   Numbers   may  be    aptly 

1  Wallis,  Opera,  I,  p.  409,  Prop.  CIV. 


ADDENDA  293 

represented  by  the  Ordinates  of  the  logarithmic 
Curve,"  continued  from  j/=— 00  away  from  the 
origin  when  dx''  is  affirmative,  or  towards  the  origin 
when  </y' ìs  negative.  In  the  "  Additions,"  p.  185, 
Craig  devotes  six  pages  to  "  An  Answer  to  Mr. 
Varignon's  Reflections  upon  Spaces  greater  than 
infinite,"  in  which  Craig  uses  the  Leibnizian  symbol 

r  five  times,  as  in  /*:  x-'dx  = .      Nowhere  in  the 

'^  '^  i-e 

hook  under  consideration   does  Craig  use  the  nota- 

tion     of    Newton.      The     "  Additions  "    are     dated 

"  September  23d,    1713." 

248.    George   Cheyne  was  a  pupil  of  the  Scotch 

physician,  Archibald  Fitcairne  (1652-17 13),  who  is 

the  author  of  two  books  on  fluxions  (which  we  bave 

not  seen),  viz.  Fluxionuni  Methodus  inversa  ;  Sive 

Quantitatum    Fluentiuin    Leges    generaliores.       Ad 

celebeì'rimum     viruni,      Archibaldum      Pitcarniuni^ 

MedicumEdinburgensem\  and  Rudimentoruin M ethodi 

Fluxionuni    inversce    Spechnina    adverstis    Abr.    de 

Moivre.       Pitcairne's    mathematica!    bent    more    or 

less  influenced  bis  medicai   theories.      He   liked  to 

ridicule  others,  and  was  himself  ridiculed  in  a  publica- 

tion,    Apollo  Mathemaiicus  ;   or,    the  Art  of  cu  ring 

Diseases    by   the    Mathematicks,    accordi ng    to    the 

Principles  of  Dr.    Pitcairne,    1695. 


INDEX 


Absolute  motion,  86,  179. 
nothing,  219,  291. 
space,  178,  179. 
"  Achilles,"  97,  125. 
Agnesi,  M.  G.,  247-250. 
Alembert,  d',  see  D'Alembert, 
Analyst    (Berkeley's),     34,    57-95, 
loi,   117,  128,   141,   142,   145, 
148,  149,   151,   153,   155,   157, 

158,  165,  179,  181,  183,  189, 
199,  229,  251,  253,  254,  263, 
266,  289. 

Analytical    Society   at   Cambridq;e, 

274. 
Angle  of  contact,  42. 
Antecedental  calculus,  235-239. 
Apollonius,  31,  155. 
Archimedes,  181,  187,  253. 
Aristotle,  166,  256. 
Arithmetisation,  225-239,  288. 

Babbage,  Charles,  271,  274. 

Barrow,  Isaac,  i,  48,  52,  65,  132, 
185  ;  bis  definition  of  velocity, 
185. 

Bayes,  Thomas,  157. 

Berkeley,  George  (Bishop),  2,  34, 
56-95,  87-91,  94-96,  loi,  112, 
123,  148,  149,  151,  158,  163, 
168,  171,  178,  183,  184,  188, 
190,  199,  200,  203,  205,  216, 
218,  222,  229,  250,  263,  265, 
266  ;  bis  first  reply  to  Jurin 
and  Walton,  72-78  ;  his  second 
reply  to  Walton,  85-87  ;  Berke- 
ley's  lemma,    59,    71,   84,    93, 

159,  160,  161,  168,  200,  265; 
Berkeley  ten  years  after,  178, 
179. 


Bernoulli,  D.,  41,  274,  278. 
John,  31,  32,  44,   114,   140,  274, 
278. 
Binomial   theorem,    60,    233,    237, 

247,  264. 
Bossut,  Ch.,  269. 
Bougainville,  L.  A.  de,  204. 
Brewster,  David,  26. 
Buffon,  Comte  de,  203-205. 
Buonaparte,  90. 

Calculus,  pre-history  of,  i  ;  contro- 
versy  on  invention  of,  38,  39, 
47,  49  ;  see  Dififerential  calculus. 

Cantabrigiensis,  209,  210,  215. 

Cantor,  Georg,  284,  288. 
Moritz,  235. 

Carnot,  L.  N.  M.,  62,  262,  263. 

Carré,  L.,  41. 

Cause  and  effect,  79,  166,  174,  185, 
192-194,  210,  212,  257, 

Cavalieri,  i,  26,  100,  116,  132. 

Cheyne,  George,  40,  41,  289-290, 

293- 
Clairaut,  A.  C,  274,  278. 
Clarke,  John,  47. 
Clerk,  G.  L.  Le,  see  Buffon,  Comte 

de. 
Collins,  John,  26,  47,  48,  112,  244. 
Colour  device  in  numbers,  137. 
Colson,   John,    149-154,    164,    165, 

182,  247,  248. 
Commerciti»!    Epistolictim,    26-29, 

36,   47-49,    52,   55,    112,   204, 

244. 
Compensation  of  errors,  62,  63,  68, 

262. 
Cote.s,  Roger,  31,  39,  55,  211. 
Cournot,  A.  A.,  94. 


294 


INDEX 


295 


Cousin,  J.  A.  J.,  269,  270. 
Craig,  John,  37-39,  55,  274,  289, 
291-293. 

Daily  Gazetteer,  212. 

D'Alembert,    204,    238,    239,    254, 

255,  257,  263,  265,  268,  271- 

273.  275,  278,  279. 
Davis,  William,  253. 
Dealtry,  William,  252. 
Dedekind,  R. ,  288. 
De  l'Hospital,  41,   50,  51,  53,   54, 

66,  84,  169,  248. 
De  Moivre,  A.,  39-41,  243. 
De  Morgan,  A.,  26,  29,  32,  34,  S^» 

39,  90,  91,  94,  238,  239,  274. 
*'  Dichotomy,"  119. 
Diderot,  D.,  275. 
Difterential  calculus,  45,  48,  52,  58, 

61,  62,  114,  118,  139,  156,  222, 

225,  239,  249,  258,  260,  261, 

271,  277,  288. 
method  of  Leibniz,  28,  29,  37,  38, 

49,217;  j^^Diff'erential  calculus. 
Ditton,  H.,  40,  41,   44,  46,  47,  50, 

53,  55-  56. 
Duhamel,  J.  M.  C,  94. 
Duillier,  Fatio  de,  39,  55. 

Edinburgh  Review.  249. 
Edleston,  J.,  31,  32,  40. 
Effect,  see  Cause  and  effect. 
Emerson,    William,    178,    192-194, 
206,  209,  213,  218,  221,  223, 

243.  259- 

Kncyclopczdia  Britaiiìiica,  240,  269, 
273,  287. 

Enestrom,  G.,  30. 

Errors,  not  to  be  neglected,  19,  23, 
28,  32,  34  ;  sec  Infinilesimals, 
Infinitely  little  quantities. 

Euclid,  7,  IO,  31,  119,  155,  184, 
•    187,  191,  222,  253. 

Euler,  L.,  139,  140,  258,  265,  274. 

Evanescent  quantities,  6,  8-10,  19, 
20,  23,  24,  26,  35,  62,  63,  68, 
70,  71,  73,  76.  77,  79,  81,  105, 

106,  116,   131,  135,   137,  156, 
162,  165-168,  191,  247,  255. 

Exhaustion,  method  of,  i,  76,  100, 

107,  128,  187,  272,  273. 


Fatio  de  Duillier,  39,  55. 

Fenn,  John,  274. 

Flowing  quantities  ;  see  Fluents. 

Fluents,  15,  16,  18,  21,  22,  25,  27- 
29,  43,  49,  55,  70,  89,  "2, 
113,  159,  162,  173,  174,  192, 
195,  199,  201,  202,  209,  211, 
213-215,  246,  273. 

Fluxions,  criticisms  of,  58,  72,  90, 
225,  229,  232,  235,  236,  255- 
276  ;  defìnition  ot — by  Emerson, 
192,  in  Encyclopcedia  Bì'i tan- 
nica, 240,  by  Maclaurin,  184, 
by  I.  Newton,  11,  13-16,  18, 
21,  22,  by  Simpson,  169,  179, 
195,  211,  213,  219,  244; 
erroneously  defined,  39-56,  71, 
156,  171,  172,  198,  240,  248; 
notation  of,  15,  16,  28-33,  3^" 
40,  43,  46,  48,  49,  54,  186, 
202,  248,  270,  273,  289-293  ; 
pre-historyof,  i  ;  second,  third, 
etc,  fluxions,  39,  42,  47,  55, 
56,  63,  74,  'jy,  80,  82,  89,  90, 
173,  182,   196,  199,  241,  272. 

Fraser,  A.  C,  178,  188. 

Function,  235,  269. 

Galileo,  I. 

Gentleman'' s  Diary,  280. 

"  Ghosts   of  departed    quantities," 

63,  68,  73,  85,  216,  250 
Gibson,  G.  A.,  93,  96. 
Glenie,  James,  235-239. 
Graves,  J.  T.,  276. 

R.   P.,  38,  90,  91,  94,    180,  274, 
276. 
Gray,  G.  J.,  165. 
Gregory,  David,  39. 

James,  48. 

St.  Vincent,  i,  99. 

Hales,  William,  267-269,  275. 

Halley,  E.,  39,  57. 

Hamilton,  W,  R.,   38,  90,  91,  94, 

180,  274,  276. 
Harris,  John,  40,  41,  55. 
Hayes,  Ch.,  41-43,  53,  55,  56. 
Heath,  Robert,  207.  210,  212,  213, 

229. 
Hellins,  John,  247-250. 


296 


LIMITS  AND  FLUXIONS 


Herschel,  J.  F.  W.,  271,  274. 
Hobhes,  Thomas,  163. 
Hodgson,  James,  155-157. 
Holliday,  F.,  243,  244. 
Hospital,  De  1',  41,  50,  51,  53,  54, 

66,  84,  169,  248. 
Hutton,  Charles,  210,  244,  245. 

Imaginary  quantities,  154. 

Imagination,  straining  of,  120-123, 
125,  147,  154,  195.  214,  217. 

Incommensurables,  7,  io. 

Indivisihles,  method  of,  5,  7,  8, 
24,  29,  100,  106,  114,  116, 
117,    127,    128,   132,  133,   139, 

150.  152,   154,   183,  204,  205, 
243- 

Infinite  divisibility,  5,  7,  io,  61,  89, 

153,  193. 

Infinite  series,  method  of,  20,  24, 
164,  221. 

Infinitely  little  quantities,  15-17, 
20,  21,  24,  26-28,  Tp-Z^,  39, 
40-43,  45,  47,  48,  50,  51,  53, 
54,  56,  58,  61,  76-78,  80,  86, 
94,  114.  117,  118,  123,  132, 
133.  137,  147,  150-156,  162, 
168,  169,  174,  182,  183,  196, 
197,  199-202,  220,  243,  248, 
258,  261,  277,  290-291  ;  avoid- 
ing  their  use,  142,  162,  169, 
171,  183,  184,  186,  189,  275  ; 
see  Infinitesimals. 

Infinitesimals,  29,  32,  35,  36,  46, 
50,  56,  58,  59,  72,  74-76,  83, 
89,  94,  I II,  1 14,  1 18,  147,  182, 
184,  201,  206.  207,  241,  250, 
263,  275,  277-279- 

Infinity,    162,  221,  249;    absolute, 

151,  163,   292;    relative,    151, 
162,  163,  183,  184,  290. 

Irrational  numbers,  2S4,  286-288. 

Jack,  Richard,  222,  223. 

Jones,  William,  17,  43,  243. 

Jourdain,  Philip  E.  B..  29,  263, 
274,  284. 

Jurm,  James,  57,  64,  72,  91,  93, 
139,  140.  189,  190,  199,  203, 
205,  206,  254,  272  ;  Jurin's 
first  reply  to   Herkeley,  64-69  ; 


second  reply,  80-85  5  contro- 
versy  with  Robins,  96-148, 179; 
controversy  with  Pemberton, 
129-138  ;  explains  history  of 
controversy  with  Robins,  145- 
146. 
Juvenal,  168. 

Keill,  John,  31,  48. 
Kelland,  Philip,  189. 
Kirkby,  John,  225-230. 
Klopstock,  188. 
Kowalewski,  G.,  17. 

Lacroix,  S.  F.,  255,  260-262,  268- 

275,  279- 

Ladies'  Diary,  207,  209,  210,  212, 
219,  221,  222,  229. 

Ladys  Philosopher,  219. 

Lagrange,  J.  L.,  94,  181,  239,  255, 
259,  266,  270-272,  274,  278  ; 
his  Fonctions  analytiqiies,  255, 
259,  263,  266,  272. 

Landen,  J.,  215,  222,  231-235,  239, 
257,  258,  261,  262,  266,  270, 
273,  280. 

Laplace,  P.  S.,  268,  278. 

Last  ratio  ;  see  Ultimate  ratio. 

Lee,  Sidney,  181,  263. 

Legendre,  A.  M.,  268,  278, 

Leibniz,  G.  W.,  i,  28,  32,  33,  37, 
38,  41,  47-49,  52,  53,  55,  94, 
106,  112-114,  118,  175,  176, 
199,  248,  254,  262,  274;  Dififer- 
ential  method  of,  see  Difieren- 
tial  calculus. 

Leibnizian  notation,  32,  ^"j,  38,  48, 
49,  53,  54,  139,  254,  274,  278, 
289,  291-293. 

Leslie,  John,  273. 

L'Huilier  (Lhuilier),  S.  A.  J,,  269. 

Limit,  I,  3-5,  7-10,  25,  33,  35,  36, 
62,  70,  75-78,  83,  89,  92,  93, 
97,  99,  10^,  I04'  ^^6,  116,  119, 
129,  146,  147,  183,  185,  189, 
191,  218,  220,  238,  241,  242, 
247,  250,  251,  254,  255,  258, 
259,  263,  265-269,  271,  272, 
275,  276,  279,  282,  286-288  ; 
limit  not  reached,  97,  102,  107, 
nò,    119,    131,    147,    186,  189, 


INDEX 


297 


221,  241,  242,  275  ;  limit  is 
reached,  97,  98,  loi,  103,  119, 
123,  124,  147,  157,  190,  271. 

Locke,  John,  69. 

Lyons,  Israel,  201. 

Maclaurin,  Colin,  181,  199,  203- 
205,  250,  253,  254,  256,  263, 
2Ó6,  268,  272,  273,  277,  283  ; 
on  infinitely  small  magnitudes, 
182  ;  li  fé  of,  183. 

Maclaurin's  Treatise  of  Fhixions, 
149,  175,  181-189,  223,  254, 
263  ;  French  translation,  189. 

Martin,  Benjamin,  171,  172,  178, 
179. 

Maseres,  Baron,  248,  267,  268. 

Matheinatical  Exercises  (Turner's), 
210,  218,  219. 

McUheniaticmn,  210. 

"  Maximinoritv,"  188. 

Méray,  H.  C.  R.,  288. 

"  Minimajority,"  188. 

Minute  Alaihemafician,  80,  136, 
.138. 

Moivre,  De,  39. 

Moments,  ri-14,  16,  25-28,  30,  31, 
33,  39,  40,  42,  46,  47,  49,  58, 
59,  66,67,  70,71,  73,  74,  76- 
y^,  89,  91,  92,  99,  100,  105, 
106,  116,  118,  123,  128,  147, 
151-155,  159,  161,  172,  176, 
183,  192,  193,  196,  197,  201  ; 
notation  for,  25,  27,  28,  31; 
origin  of  term,  26. 

Moment  of  a  rectangle,  12,  13,  58, 
66,  71,  74,  75,  79.82,  83,  91, 
92,  151- 

Monthly  Revteiv,  200,  209,  210, 
215,  222,  224,  234,  255,  259- 
262,  267,  268. 

Morgan,  De  ;  sec  De  Morgan,  A. 

Morgan,  James,  210. 

Motte,  A.,  translator  of  Principia^ 
4,  12,  103. 

Muller,  John,  162-164. 

Napier,  J.,  i. 

Nascent  magnitudes,  11,  13,  22,  43, 
44,  58,  59,  62,  66,  70,  73,  74, 
77,  81,  84,  io5,  113,   116,  118, 


126,  135,  156,  161,  162,  165- 
167,   172,  177,   191,   194,  243, 

245- 

Navier,  L.  M.  H.,  94. 

Newton,  Isaac,  2-36,  38-41,  43, 
49,  52,  53,  55,  56,61,  64,  65, 
67-70,  75,  78,  81-86,  S)^,  91, 
92,  96,  100,  102,  104,  111-114, 
118,    122,   123,  127  129,    130, 

133,  m,  141-143.  147,  152, 

154,  155.  157,  158,   160,  i6i, 

164,  165,  168,  170-173,  175, 
181-183,  191,  192,  195,  198, 
203-205,  211,  214,  219,  220, 
229,  235,  241,  244,  248,  252, 
254,  256,  257,  261,  264,  268, 
270, 272. 

Newton's  Analysis  per  (eqiiaiiones, 

17,  36,  48,  190,  204. 

lemmas,    3,    43,    75-77,    83,   91, 

101-103,   105,    107,   109,    110, 

115,   118,    124,   128,    131,   132, 

134,  135,  '^yi^  138,  145,  146, 
151,  158,  189,  190, 203,  241, 
242, 246. 250, 254. 

manuscripts,  29. 

Mcthod  of  Fliixions,  149,  164, 
248. 

notation  for  fìuxions  ;  see  Fluxions, 
notation  of. 

Opticks,  17,  127,  204. 

Principia,  quotations  from,  2-14, 
43  ;  see  Principia  (Newton's). 

Quadratura  Ciirvarmn  ;  see  Quad- 
ratura Curvaruvi. 

use  of  infinitely  little  quantities, 

29,  30,  32,  33-36,  155- 
Newton,  T. ,  250,  251. 
Nieuwentiit,  B.,  41,  45,  46,  102. 
Notation  for  fluxions,  see  Fluxions, 

notation    of  ;    for    calculus,   see 

Leibnizian  notation. 
Nothing,  absolute,   219,    290-291  ; 

relative,  290-291. 

Oldenburgh,  26. 
Ostwald's  Klassiker,  17. 

Palladium,  210,  212,  219. 
Parent,  A.,  I14. 
Pascal,  B.,  132. 


298 


LIMITS  AND  FLUXIONS 


Peacock,  George,  271,  272,  274. 

Pemberton,  II.,  96,  125,  129-138, 
147,  1Ó5. 

Petvin,  John,  230. 

Pezenas,  Esprit,  189. 

"  Philalethes  Cantabrigiensis,"  64, 
72,  82,  loi,  106,  108-110,  117, 
122,  124-128,  133,  136-138, 
140-144,  181,  199,  204,  205; 
see  Jurin,  James. 

"  Phileleutherus  Oxoniensis,"  133. 

Philosophicai  Transactions,  26,  38- 
41,  112,  118,  157,  185,  235, 
244,  274. 

Pitcairne,  Archibald,  293. 

Poisson,  S.  D,,  94. 

Portsmouth  collection,  30. 

Present  State  of  the  Kepiiblick  of 
Letters,  99,  loi,  106,  109,  in, 
117,  123-125,  129,  130,  133, 
135,  142,  143,  154,  162. 

Price,  B.,  278. 

Prime  and  ultimate  ratios,  5,  6,  8, 
19,  20,  22,  24,  29,  33,  35,  36, 
62,  71.  80,  81,  86,  88,  89,  96, 
100,  109,  113,  114,  117,  118, 
122,  124,  126,  128,  130,  141, 
143,  147,  166,  186,  190,  241, 
243'  250,  258.  265,  267,  273. 

Principia  (Newton's),  28,  30,  31 - 
33,  36,  53,  67-69,  74-77,  86, 
101-104,  107,  no,  126,  128, 
129,  135,  I5^>  190,  219,  241- 
243,  253  ;  quotations  from,  2- 
•4,  43- 

Quadratura  Curvarum  (Newton's), 
17-26,  28,  31-34,  36,  43,  50. 
53,  67,  69,  71,  74,  76,  77,  112, 
124,  130,  135-137,  156,  179, 
190,  246. 

Quadrature  of  curves  ;  see  Quadra- 
tura Curvarum. 


Raphson,  Joseph,  49,  55 

Ratios,    first   and    last  ;    see    Prime 

and  ultimate  ratios. 
Relative      infinite  ;      see      Infinity, 

relative. 
Relative  motion,  86. 


Relative     nothing  ;     see     Nothing, 

relative. 
Republick  of  Letters  ;    see   Present 

State  o/the  Republick  of  Letters. 
Residuai  analysis  (Landen's),  232- 

235,  239,  258,  273. 
Rigaud,  S,  P.,  14,  29,  30. 
Robartes,  F.,  41. 
Roberval,  i. 
Robins,  B.,  96-148,  179,  181,    188, 

189,    199,  242,   250,  254,  2Ó6, 

267,  272-275  ;  Robins'sil/a/A^- 

matical       Tracts,        202-206  ; 

Robins's     Discourse,    96-100, 

115,  141,    142,  267,  268,  275; 

explains  history  of  conlroversy 

with  Jurin,  141-145. 
Rowe,  John,    172,    173,    175,    178, 

179,  195,  202,  253,  283. 
Rowning,  John,  198-200. 

Saunderson,  Nicholas,  150,  197, 
198,  204,  242. 

Simpson,  Thomas,  169-171,  175, 
178-180,  194-196,  206,  209, 
210-215,  218,  220,  221,  223, 
224,  244,  253,  259,  277  ;  his 
definition  of  fluxions.  244, 

Simson,  Robert,  280. 

Sloane,  H.,  48. 

Smith,  James,  165-169. 

Smith,  Robert,  139,  140,  201. 

Solidus,  use  of,  12. 

Stevin,  S. ,  99. 

Stewart,  John,  17,  21.  25,  190-192. 

Stirling,  James,  50,  55. 

Stockler,  G.,  259. 

Stona,  E.,  50.  51,  53-56,  169,  248, 
259. 

St.  Vincent,  Gregory,  i,  99. 

Stubbs,  J.,  177. 

Tacquet,  Andrew,  99. 

Tannery,  Jules,  137. 

Taylor,  Brook,   50. 

Thorp,  R.,  241-243;  translator  of 

Principia,  3,  8. 
Torelli,  G.,  257. 
Truth   Triumphant,  212,  213,  215, 

218,  222,  223. 
Turner,  John,  210-214,  218,  219. 


INDEX 


299 


Ultimate  ratios,  6,  7,  9,  io,  19,  22- 
24,  35,  68,  70,  78,  86,  98,  99, 
103,  115,  153,  158-161,  168, 
191,  219,  223,  242. 

Ultimate  velocity,  7,  io;  see 
Velocity. 

Ultimato! s,  doctrine  of.  225. 

Valerius,  Lucas,  99. 

Varignon,  P. ,  293. 

Velocity,  II,  13-18,  21,  28-30,  32, 
33,  36,  43-45,  49,  50,  58,  62, 
63,  66,  70,  71,  74,  76-80,  83, 
85.  87,  88,  93,  106,  112,  123, 
151,  152,  158,  163,  164,  166, 
169-171,  173,  175,  184,  185, 
193-197,  I99>  200,  201,  207, 
209,  212,  214,  215,  217,  219, 
220,  231,  235,  244,  246,  252, 
261,  263,  270,  277,  280  ;  velo- 
city,  ultimate,  6,  9,  36  ;  velo- 
city  criticised  as  a  fundamental 
concept,  255,  264,  280-282. 

Vince,  S.,  245-247,  277, 

Vivanti,  G.,  235. 


Wallis,  John,  41,  100,  292  ;  Wallis's 
Algebra,  14,  30,  32,  33,  39. 

Walton,  John,  57,  69,  77,  85,  86, 
91-96,  148,  185,  199;  his  first 
reply  to  Berkeley,  69-72  ;  his 
second  reply,  78-80  ;  second 
edition  of  second  reply,  87. 

Warburton,  Bishop  William,  ']'},. 

Waring,  E.,  259. 

Weierstrass,  K..  288. 

Weissenborn,  H.,  92. 

West,  William,  202. 

Whately,  90. 

Whiston,  William,  197. 

Wildbore,  280. 

Wilson,  James,  96,  99,  189,  202- 
206, 

Witting,  A.,  30. 

Wood,  James,  245, 

Woodhouse,  R.,  93,  263-267,  274, 
280. 

Works  of  the  Learned,  135-137, 
171. 

Zeno  (of  Elea),  90,  119,  125,  217. 


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